3358:
2563:
3353:{\displaystyle {\begin{aligned}4&=2^{2}3^{0},&6&=2^{1}3^{1},&\gcd(4,6)&=2^{1}3^{0}=2,&\operatorname {lcm} (4,6)&=2^{2}3^{1}=12.\\{\tfrac {1}{3}}&=2^{0}3^{-1}5^{0},&{\tfrac {2}{5}}&=2^{1}3^{0}5^{-1},&\gcd \left({\tfrac {1}{3}},{\tfrac {2}{5}}\right)&=2^{0}3^{-1}5^{-1}={\tfrac {1}{15}},&\operatorname {lcm} \left({\tfrac {1}{3}},{\tfrac {2}{5}}\right)&=2^{1}3^{0}5^{0}=2,\\{\tfrac {1}{6}}&=2^{-1}3^{-1},&{\tfrac {3}{4}}&=2^{-2}3^{1},&\gcd \left({\tfrac {1}{6}},{\tfrac {3}{4}}\right)&=2^{-2}3^{-1}={\tfrac {1}{12}},&\operatorname {lcm} \left({\tfrac {1}{6}},{\tfrac {3}{4}}\right)&=2^{-1}3^{1}={\tfrac {3}{2}}.\end{aligned}}}
31:
5515:
1922:
5484:
191:
of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5
4576:
4953:
715:
teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete to realign the line segment can be calculated by using
3688:
1512:
4286:
1335:
1765:
1878:
4397:
1943:(gcd), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the gcd of 48 and 180 is 2 × 2 × 3 = 12.
2092:
686:
4408:
4717:
2401:
406:
3423:
given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a
1640:
4796:
3514:
1176:
2568:
2302:
4100:
2552:
510:
3918:
3855:
3778:
1669:
1067:
1018:
969:
838:
795:
1533:. As these algorithms are more efficient with factors of similar size, it is more efficient to divide the largest argument of the lcm by the gcd of the arguments, as in the example above.
3981:
3592:
920:
1770:
The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 2, 3, and 7, respectively. Thus,
4787:
752:
260:
4159:
2478:
1406:
2557:
In fact, every rational number can be written uniquely as the product of primes, if negative exponents are allowed. When this is done, the above formulas remain valid. For example:
581:
3574:
4178:
2212:
2162:
5081:
In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are
1187:
4026:
1664:
1776:
5460:
4292:
3433:
If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true.
878:
are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after
4571:{\displaystyle \gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (b,c),\operatorname {lcm} (a,c))=\operatorname {lcm} (\gcd(a,b),\gcd(b,c),\gcd(a,c)).}
1970:
612:
4617:
1645:
Here, the composite number 90 is made up of one atom of the prime number 2, two atoms of the prime number 3, and one atom of the prime number 5.
2313:
271:
4948:{\displaystyle \operatorname {lcm} (a_{1},a_{2},\ldots ,a_{r})=\operatorname {lcm} (\operatorname {lcm} (a_{1},a_{2},\ldots ,a_{r-1}),a_{r}).}
5381:
5356:
5313:
1562:
3454:
1095:
1529:. For very large integers, there are even faster algorithms for the three involved operations (multiplication, gcd, and division); see
5453:
5338:
1957:
1902:
factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram.
1883:
This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for
1542:
5330:
2220:
4044:
5672:
5667:
2489:
417:
5446:
3861:
3798:
3694:
1023:
974:
925:
800:
757:
5086:
3683:{\displaystyle \operatorname {lcm} (a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\operatorname {lcm} (a,b),c),}
30:
3939:
5626:
1507:{\displaystyle \operatorname {lcm} (21,6)=6\times {\frac {21}{\gcd(21,6)}}=6\times {\frac {21}{3}}=6\times 7=42.}
881:
603:
129:
4733:
1181:
To avoid introducing integers that are larger than the result, it is convenient to use the equivalent formulas
719:
227:
4106:
2412:
5614:
5404:
5373:
4281:{\displaystyle \operatorname {lcm} (a,\gcd(b,c))=\gcd(\operatorname {lcm} (a,b),\operatorname {lcm} (a,c)),}
1940:
1086:
606:) is used, because each of the fractions can be expressed as a fraction with this denominator. For example,
524:
5677:
5496:
5118:
5090:
3520:
3425:
184:
125:
3447:
1884:
1526:
188:
2167:
2117:
1330:{\displaystyle \operatorname {lcm} (a,b)=|a|\,{\frac {|b|}{\gcd(a,b)}}=|b|\,{\frac {|a|}{\gcd(a,b)}},}
5609:
4169:
1545:
indicates that every positive integer greater than 1 can be written in only one way as a product of
1895:
1655:
1530:
1522:
3987:
1760:{\displaystyle {\begin{aligned}8&=2^{3}\\9&=3^{2}\\21&=3^{1}\cdot 7^{1}\end{aligned}}}
5572:
5557:
5128:
5082:
3420:
3416:
849:
3439:
The following pairs of dual formulas are special cases of general lattice-theoretic identities.
5587:
5428:
5408:
5377:
5352:
5334:
5309:
3369:
1873:{\displaystyle \operatorname {lcm} (8,9,21)=2^{3}\cdot 3^{2}\cdot 7^{1}=8\cdot 9\cdot 7=504.}
1549:. The prime numbers can be considered as the atomic elements which, when combined, make up a
5631:
5123:
4964:
4392:{\displaystyle \gcd(a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\gcd(a,b),\gcd(a,c)).}
3385:
1550:
1390:
98:
5621:
5582:
4037:
3412:
1960:, every integer greater than 1 can be represented uniquely as a product of prime numbers,
599:
5154:
691:
where the denominator 42 was used, because it is the least common multiple of 21 and 6.
3789:
5058:
is a common multiple that is minimal, in the sense that for any other common multiple
1921:
5661:
5646:
5567:
5424:
5214:
3585:
47:
2214:, their least common multiple and greatest common divisor are given by the formulas
922:
units of time. At this time, the first, second and third planet will have completed
5597:
5592:
5562:
5101:
can be characterised as a generator of the intersection of the ideals generated by
1891:
1546:
35:
2087:{\displaystyle n=2^{n_{2}}3^{n_{3}}5^{n_{5}}7^{n_{7}}\cdots =\prod _{p}p^{n_{p}},}
5367:
5324:
4723:
681:{\displaystyle {2 \over 21}+{1 \over 6}={4 \over 42}+{7 \over 42}={11 \over 42}}
17:
5641:
5636:
4604:
3932:
43:
132:" (lcd), and can be used for adding, subtracting or comparing the fractions.
5483:
1518:
797:
rotations for the realignment. By that time, the second gear will have made
86:
4712:{\displaystyle |\{(x,y)\;:\;\operatorname {lcm} (x,y)=D\}|=3^{\omega (D)},}
3408:). (Note that the usual magnitude-based definition of ≤ is not used here.)
163:, is defined as the smallest positive integer that is divisible by each of
2396:{\displaystyle \operatorname {lcm} (a,b)=\prod _{p}p^{\max(a_{p},b_{p})}.}
401:{\displaystyle 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,...}
5504:
5469:
5604:
5577:
5500:
5438:
704:
63:
1635:{\displaystyle 90=2^{1}\cdot 3^{2}\cdot 5^{1}=2\cdot 3\cdot 3\cdot 5.}
2111:, ... are non-negative integers; for example, 84 = 2 3 5 7 11 13 ...
38:
showing the least common multiples of all subsets of {2, 3, 4, 5, 7}.
3509:{\displaystyle \operatorname {lcm} (a,b)=\operatorname {lcm} (b,a),}
1171:{\displaystyle \operatorname {lcm} (a,b)={\frac {|ab|}{\gcd(a,b)}}.}
866:
units of time, respectively, to complete their orbits. Assume that
854:
Suppose there are three planets revolving around a star which take
5473:
4172:; that is, lcm distributes over gcd and gcd distributes over lcm:
1961:
602:, the least common multiple of the denominators (often called the
5412:
5109:(the intersection of a collection of ideals is always an ideal).
5432:
700:
109:
are both different from zero. However, some authors define lcm(
5442:
1654:
Factor each number and express it as a product of prime number
1525:
for computing the gcd that do not require the numbers to be
3926:
3441:
1898:
of each of the two numbers demonstrated in each circle and
1648:
This fact can be used to find the lcm of a set of numbers.
5230:
The next three formulas are from Landau, Ex. III.3, p. 254
1077:
There are several ways to compute least common multiples.
2297:{\displaystyle \gcd(a,b)=\prod _{p}p^{\min(a_{p},b_{p})}}
5369:
4963:
The least common multiple can be defined generally over
4095:{\displaystyle a\geq b\iff a=\operatorname {lcm} (a,b),}
5271:
5269:
1929:
Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
1340:
where the result of the division is always an integer.
5089:, any two elements have a least common multiple. In a
3332:
3282:
3267:
3239:
3186:
3171:
3116:
3066:
2999:
2984:
2956:
2893:
2878:
2813:
2756:
2547:{\displaystyle \gcd(a,b)\operatorname {lcm} (a,b)=ab.}
2170:
2120:
1935:
Product = 2 × 2 × 2 × 2 × 3 × 2 × 2 × 3 × 3 × 5 = 8640
505:{\displaystyle 6,12,18,24,30,36,42,48,54,60,66,72,...}
4799:
4736:
4620:
4411:
4295:
4181:
4109:
4047:
3990:
3942:
3864:
3801:
3697:
3595:
3523:
3457:
2566:
2492:
2415:
2316:
2223:
1973:
1779:
1667:
1565:
1409:
1190:
1098:
1027:
978:
929:
884:
804:
761:
722:
615:
527:
420:
274:
230:
124:
The least common multiple of the denominators of two
3913:{\displaystyle \gcd(a,\operatorname {lcm} (a,b))=a.}
3850:{\displaystyle \operatorname {lcm} (a,\gcd(a,b))=a,}
3773:{\displaystyle \gcd(a,\gcd(b,c))=\gcd(\gcd(a,b),c).}
3411:
Under this ordering, the positive integers become a
586:
In this list, the smallest number is 12. Hence, the
135:
The least common multiple of more than two integers
5522:
5489:
5419:Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970),
1085:The least common multiple can be computed from the
1062:{\displaystyle \operatorname {lcm} (l,m,n) \over n}
1013:{\displaystyle \operatorname {lcm} (l,m,n) \over m}
964:{\displaystyle \operatorname {lcm} (l,m,n) \over l}
518:of 4 and 6 are the numbers that are in both lists:
4947:
4781:
4711:
4570:
4391:
4280:
4153:
4094:
4020:
3975:
3912:
3849:
3772:
3682:
3568:
3508:
3352:
2546:
2472:
2395:
2296:
2206:
2156:
2086:
1872:
1759:
1634:
1506:
1343:These formulas are also valid when exactly one of
1329:
1170:
1061:
1012:
963:
914:
832:
789:
746:
680:
575:
504:
400:
254:
101:is undefined, this definition has meaning only if
27:Smallest positive number divisible by two integers
5260:
5201:
833:{\displaystyle \operatorname {lcm} (m,n) \over n}
790:{\displaystyle \operatorname {lcm} (m,n) \over m}
4544:
4523:
4502:
4412:
4365:
4344:
4296:
4221:
4197:
4130:
3991:
3865:
3817:
3740:
3734:
3710:
3698:
3545:
3524:
3162:
2869:
2641:
2493:
2437:
2416:
2356:
2260:
2224:
1446:
1303:
1247:
1144:
1890:The same method can also be illustrated with a
5454:
8:
4676:
4626:
3976:{\displaystyle \operatorname {lcm} (a,a)=a,}
5323:Crandall, Richard; Pomerance, Carl (2001),
915:{\displaystyle \operatorname {lcm} (l,m,n)}
85:, is the smallest positive integer that is
5461:
5447:
5439:
5326:Prime Numbers: A Computational Perspective
5239:Crandall & Pomerance, ex. 2.4, p. 101.
4648:
4644:
4168:It can also be shown that this lattice is
4123:
4119:
4061:
4057:
200:The least common multiple of two integers
4933:
4911:
4892:
4879:
4845:
4826:
4813:
4798:
4782:{\displaystyle a_{1},a_{2},\ldots ,a_{r}}
4773:
4754:
4741:
4735:
4691:
4679:
4621:
4619:
4410:
4294:
4180:
4108:
4046:
3989:
3941:
3863:
3800:
3696:
3594:
3522:
3456:
3331:
3322:
3309:
3281:
3266:
3238:
3226:
3213:
3185:
3170:
3151:
3138:
3115:
3101:
3088:
3065:
3046:
3036:
3026:
2998:
2983:
2955:
2943:
2930:
2920:
2892:
2877:
2855:
2845:
2835:
2812:
2801:
2788:
2778:
2755:
2739:
2729:
2680:
2670:
2630:
2620:
2595:
2585:
2567:
2565:
2491:
2414:
2379:
2366:
2355:
2345:
2315:
2283:
2270:
2259:
2249:
2222:
2196:
2191:
2181:
2169:
2146:
2141:
2131:
2119:
2073:
2068:
2058:
2040:
2035:
2023:
2018:
2006:
2001:
1989:
1984:
1972:
1840:
1827:
1814:
1778:
1747:
1734:
1710:
1686:
1668:
1666:
1602:
1589:
1576:
1564:
1476:
1440:
1408:
1296:
1288:
1285:
1284:
1279:
1271:
1240:
1232:
1229:
1228:
1223:
1215:
1189:
1137:
1126:
1123:
1097:
1025:
976:
927:
883:
802:
759:
747:{\displaystyle \operatorname {lcm} (m,n)}
721:
668:
655:
642:
629:
616:
614:
526:
419:
273:
255:{\displaystyle \operatorname {lcm} (4,6)}
229:
117:, since 0 is the only common multiple of
5401:Elementary Introduction to Number Theory
5351:(2nd ed.). New York, NY: Springer.
4154:{\displaystyle a\leq b\iff a=\gcd(a,b).}
2473:{\displaystyle \min(x,y)+\max(x,y)=x+y,}
1932:Greatest common divisor = 2 × 2 × 3 = 12
29:
5287:
5140:
1069:orbits, respectively, around the star.
598:When adding, subtracting, or comparing
5275:
5185:
5183:
4722:where the absolute bars || denote the
4038:Define divides in terms of lcm and gcd
1916:sharing two "2"s and a "3" in common:
1397:must be considered as a special case.
576:{\displaystyle 12,24,36,48,60,72,...}
7:
5366:Hardy, G. H.; Wright, E. M. (1979),
5248:
5189:
5148:
5146:
5144:
3569:{\displaystyle \gcd(a,b)=\gcd(b,a).}
3435:(Remember ≤ is defined as divides).
4599:) distinct prime numbers (that is,
5306:A First Course in Rings and Ideals
4978:be elements of a commutative ring
2207:{\textstyle b=\prod _{p}p^{b_{p}}}
2157:{\textstyle a=\prod _{p}p^{a_{p}}}
25:
1958:fundamental theorem of arithmetic
1952:Fundamental theorem of arithmetic
1081:Using the greatest common divisor
5513:
5482:
5347:Grillet, Pierre Antoine (2007).
5177:Hardy & Wright, § 5.1, p. 48
1920:
1400:To return to the example above,
5308:. Reading, MA: Addison-Wesley.
5093:, the least common multiple of
5014:(that is, there exist elements
754:. The first gear must complete
5261:Pettofrezzo & Byrkit (1970
5202:Pettofrezzo & Byrkit (1970
4939:
4923:
4872:
4863:
4851:
4806:
4701:
4695:
4680:
4667:
4655:
4641:
4629:
4622:
4562:
4559:
4547:
4538:
4526:
4517:
4505:
4499:
4487:
4484:
4472:
4460:
4448:
4436:
4424:
4415:
4383:
4380:
4368:
4359:
4347:
4341:
4329:
4326:
4314:
4299:
4272:
4269:
4257:
4245:
4233:
4224:
4215:
4212:
4200:
4188:
4145:
4133:
4120:
4086:
4074:
4058:
4006:
3994:
3961:
3949:
3898:
3895:
3883:
3868:
3835:
3832:
3820:
3808:
3764:
3755:
3743:
3737:
3728:
3725:
3713:
3701:
3674:
3665:
3653:
3644:
3632:
3629:
3617:
3602:
3560:
3548:
3539:
3527:
3500:
3488:
3476:
3464:
2715:
2703:
2656:
2644:
2529:
2517:
2508:
2496:
2452:
2440:
2431:
2419:
2385:
2359:
2335:
2323:
2289:
2263:
2239:
2227:
1804:
1786:
1461:
1449:
1428:
1416:
1318:
1306:
1297:
1289:
1280:
1272:
1262:
1250:
1241:
1233:
1224:
1216:
1209:
1197:
1159:
1147:
1138:
1127:
1117:
1105:
1052:
1034:
1003:
985:
954:
936:
909:
891:
823:
811:
780:
768:
741:
729:
249:
237:
216:). Some older textbooks use .
1:
3368:The positive integers may be
1389:, these formulas would cause
147:, . . . , usually denoted by
4402:This identity is self-dual:
4021:{\displaystyle \gcd(a,a)=a.}
2114:Given two positive integers
1543:unique factorization theorem
99:division of integers by zero
5403:(2nd ed.), Lexington:
5087:unique factorization domain
5694:
1964:the order of the factors:
847:
5553:
5511:
5480:
5421:Elements of Number Theory
5304:Burton, David M. (1970).
4032:
3784:
3580:
1537:Using prime factorization
604:lowest common denominator
130:lowest common denominator
5399:Long, Calvin T. (1972),
5392:Elementary Number Theory
1939:This also works for the
1912:180 = 2 × 2 × 3 × 3 × 5,
60:smallest common multiple
5405:D. C. Heath and Company
5390:Landau, Edmund (1966),
5374:Oxford University Press
5155:"Least Common Multiple"
1941:greatest common divisor
1909:48 = 2 × 2 × 2 × 2 × 3,
1089:(gcd) with the formula
1087:greatest common divisor
5119:Anomalous cancellation
5091:principal ideal domain
4949:
4783:
4713:
4572:
4393:
4282:
4155:
4096:
4022:
3977:
3914:
3851:
3774:
3684:
3570:
3510:
3354:
2548:
2474:
2397:
2298:
2208:
2158:
2088:
1874:
1761:
1636:
1508:
1331:
1172:
1063:
1014:
965:
916:
834:
791:
748:
699:Suppose there are two
682:
577:
506:
402:
256:
56:lowest common multiple
39:
5673:Operations on numbers
5668:Elementary arithmetic
5159:mathworld.wolfram.com
5048:least common multiple
4950:
4784:
4714:
4573:
4394:
4283:
4156:
4097:
4023:
3978:
3915:
3852:
3775:
3685:
3571:
3511:
3419:given by the gcd and
3355:
2549:
2475:
2398:
2299:
2209:
2159:
2089:
1894:as follows, with the
1885:integer factorization
1875:
1762:
1651:Example: lcm(8,9,21)
1637:
1509:
1332:
1173:
1064:
1015:
966:
917:
835:
792:
749:
683:
588:least common multiple
578:
507:
403:
257:
73:, usually denoted by
52:least common multiple
33:
5423:, Englewood Cliffs:
4959:In commutative rings
4797:
4734:
4618:
4409:
4293:
4179:
4107:
4045:
3988:
3940:
3862:
3799:
3695:
3593:
3521:
3455:
3372:by divisibility: if
2564:
2490:
2413:
2314:
2221:
2168:
2118:
2097:where the exponents
1971:
1905:Here is an example:
1777:
1665:
1563:
1407:
1188:
1096:
1024:
975:
926:
882:
801:
758:
720:
613:
525:
418:
411:Multiples of 6 are:
272:
265:Multiples of 4 are:
228:
5394:, New York: Chelsea
5153:Weisstein, Eric W.
1896:prime factorization
1531:Fast multiplication
1523:Euclidean algorithm
1373:. However, if both
844:Planetary alignment
187:of a number is the
5490:Division and ratio
5129:Chebyshev function
4945:
4779:
4709:
4591:be the product of
4568:
4389:
4278:
4151:
4092:
4018:
3973:
3910:
3847:
3770:
3680:
3566:
3506:
3400:(or equivalently,
3350:
3348:
3341:
3291:
3276:
3248:
3195:
3180:
3125:
3075:
3008:
2993:
2965:
2902:
2887:
2822:
2765:
2544:
2470:
2393:
2350:
2294:
2254:
2204:
2186:
2154:
2136:
2084:
2063:
1870:
1757:
1755:
1632:
1504:
1327:
1168:
1055:
1006:
957:
912:
850:Syzygy (astronomy)
826:
783:
744:
678:
573:
502:
398:
252:
208:is denoted as lcm(
113:, 0) as 0 for all
40:
5655:
5654:
5383:978-0-19-853171-5
5358:978-0-387-71568-1
5315:978-0-201-00731-2
4965:commutative rings
4166:
4165:
3925:
3924:
3370:partially ordered
3364:Lattice-theoretic
3340:
3290:
3275:
3247:
3194:
3179:
3124:
3074:
3007:
2992:
2964:
2901:
2886:
2821:
2764:
2341:
2245:
2177:
2127:
2054:
1956:According to the
1484:
1465:
1322:
1266:
1163:
1059:
1010:
961:
830:
787:
676:
663:
650:
637:
624:
16:(Redirected from
5685:
5632:Musical interval
5545:
5544:
5542:
5541:
5538:
5535:
5517:
5516:
5486:
5463:
5456:
5449:
5440:
5435:
5415:
5395:
5386:
5362:
5349:Abstract Algebra
5343:
5319:
5291:
5285:
5279:
5273:
5264:
5258:
5252:
5246:
5240:
5237:
5231:
5228:
5222:
5221:
5219:
5215:"nasa spacemath"
5211:
5205:
5199:
5193:
5187:
5178:
5175:
5169:
5168:
5166:
5165:
5150:
5124:Coprime integers
5108:
5104:
5100:
5096:
5077:
5073:
5069:
5065:
5061:
5057:
5053:
5045:
5035:
5025:
5021:
5017:
5013:
5009:
5005:
5001:
4997:
4993:
4989:
4981:
4977:
4973:
4954:
4952:
4951:
4946:
4938:
4937:
4922:
4921:
4897:
4896:
4884:
4883:
4850:
4849:
4831:
4830:
4818:
4817:
4788:
4786:
4785:
4780:
4778:
4777:
4759:
4758:
4746:
4745:
4718:
4716:
4715:
4710:
4705:
4704:
4683:
4625:
4577:
4575:
4574:
4569:
4398:
4396:
4395:
4390:
4287:
4285:
4284:
4279:
4160:
4158:
4157:
4152:
4101:
4099:
4098:
4093:
4027:
4025:
4024:
4019:
3982:
3980:
3979:
3974:
3927:
3919:
3917:
3916:
3911:
3856:
3854:
3853:
3848:
3779:
3777:
3776:
3771:
3689:
3687:
3686:
3681:
3586:Associative laws
3575:
3573:
3572:
3567:
3515:
3513:
3512:
3507:
3448:Commutative laws
3442:
3386:integer multiple
3359:
3357:
3356:
3351:
3349:
3342:
3333:
3327:
3326:
3317:
3316:
3297:
3293:
3292:
3283:
3277:
3268:
3249:
3240:
3234:
3233:
3221:
3220:
3201:
3197:
3196:
3187:
3181:
3172:
3156:
3155:
3146:
3145:
3126:
3117:
3109:
3108:
3096:
3095:
3076:
3067:
3051:
3050:
3041:
3040:
3031:
3030:
3014:
3010:
3009:
3000:
2994:
2985:
2966:
2957:
2951:
2950:
2938:
2937:
2925:
2924:
2908:
2904:
2903:
2894:
2888:
2879:
2863:
2862:
2850:
2849:
2840:
2839:
2823:
2814:
2806:
2805:
2796:
2795:
2783:
2782:
2766:
2757:
2744:
2743:
2734:
2733:
2685:
2684:
2675:
2674:
2635:
2634:
2625:
2624:
2600:
2599:
2590:
2589:
2553:
2551:
2550:
2545:
2479:
2477:
2476:
2471:
2402:
2400:
2399:
2394:
2389:
2388:
2384:
2383:
2371:
2370:
2349:
2303:
2301:
2300:
2295:
2293:
2292:
2288:
2287:
2275:
2274:
2253:
2213:
2211:
2210:
2205:
2203:
2202:
2201:
2200:
2185:
2163:
2161:
2160:
2155:
2153:
2152:
2151:
2150:
2135:
2093:
2091:
2090:
2085:
2080:
2079:
2078:
2077:
2062:
2047:
2046:
2045:
2044:
2030:
2029:
2028:
2027:
2013:
2012:
2011:
2010:
1996:
1995:
1994:
1993:
1924:
1879:
1877:
1876:
1871:
1845:
1844:
1832:
1831:
1819:
1818:
1766:
1764:
1763:
1758:
1756:
1752:
1751:
1739:
1738:
1715:
1714:
1691:
1690:
1641:
1639:
1638:
1633:
1607:
1606:
1594:
1593:
1581:
1580:
1551:composite number
1513:
1511:
1510:
1505:
1485:
1477:
1466:
1464:
1441:
1396:
1391:division by zero
1388:
1384:
1378:
1372:
1370:
1358:
1354:
1348:
1336:
1334:
1333:
1328:
1323:
1321:
1301:
1300:
1292:
1286:
1283:
1275:
1267:
1265:
1245:
1244:
1236:
1230:
1227:
1219:
1177:
1175:
1174:
1169:
1164:
1162:
1142:
1141:
1130:
1124:
1068:
1066:
1065:
1060:
1026:
1019:
1017:
1016:
1011:
977:
970:
968:
967:
962:
928:
921:
919:
918:
913:
839:
837:
836:
831:
803:
796:
794:
793:
788:
760:
753:
751:
750:
745:
687:
685:
684:
679:
677:
669:
664:
656:
651:
643:
638:
630:
625:
617:
600:simple fractions
582:
580:
579:
574:
516:Common multiples
511:
509:
508:
503:
407:
405:
404:
399:
261:
259:
258:
253:
192:and −2 as well.
162:
84:
21:
5693:
5692:
5688:
5687:
5686:
5684:
5683:
5682:
5658:
5657:
5656:
5651:
5622:Just intonation
5549:
5539:
5536:
5533:
5532:
5530:
5529:
5518:
5514:
5509:
5487:
5476:
5467:
5418:
5398:
5389:
5384:
5365:
5359:
5346:
5341:
5322:
5316:
5303:
5300:
5295:
5294:
5286:
5282:
5274:
5267:
5259:
5255:
5247:
5243:
5238:
5234:
5229:
5225:
5217:
5213:
5212:
5208:
5200:
5196:
5188:
5181:
5176:
5172:
5163:
5161:
5152:
5151:
5142:
5137:
5115:
5106:
5102:
5098:
5094:
5075:
5071:
5067:
5063:
5059:
5055:
5051:
5037:
5027:
5023:
5019:
5015:
5011:
5007:
5003:
5002:such that both
4999:
4995:
4991:
4987:
4984:common multiple
4979:
4975:
4971:
4961:
4929:
4907:
4888:
4875:
4841:
4822:
4809:
4795:
4794:
4769:
4750:
4737:
4732:
4731:
4687:
4616:
4615:
4584:
4407:
4406:
4291:
4290:
4177:
4176:
4105:
4104:
4043:
4042:
3986:
3985:
3938:
3937:
3933:Idempotent laws
3860:
3859:
3797:
3796:
3790:Absorption laws
3693:
3692:
3591:
3590:
3519:
3518:
3453:
3452:
3366:
3347:
3346:
3318:
3305:
3298:
3265:
3261:
3253:
3222:
3209:
3202:
3169:
3165:
3160:
3147:
3134:
3127:
3113:
3097:
3084:
3077:
3062:
3061:
3042:
3032:
3022:
3015:
2982:
2978:
2970:
2939:
2926:
2916:
2909:
2876:
2872:
2867:
2851:
2841:
2831:
2824:
2810:
2797:
2784:
2774:
2767:
2752:
2751:
2735:
2725:
2718:
2695:
2676:
2666:
2659:
2639:
2626:
2616:
2609:
2604:
2591:
2581:
2574:
2562:
2561:
2488:
2487:
2411:
2410:
2375:
2362:
2351:
2312:
2311:
2279:
2266:
2255:
2219:
2218:
2192:
2187:
2166:
2165:
2142:
2137:
2116:
2115:
2110:
2103:
2069:
2064:
2036:
2031:
2019:
2014:
2002:
1997:
1985:
1980:
1969:
1968:
1954:
1949:
1836:
1823:
1810:
1775:
1774:
1754:
1753:
1743:
1730:
1723:
1717:
1716:
1706:
1699:
1693:
1692:
1682:
1675:
1663:
1662:
1598:
1585:
1572:
1561:
1560:
1539:
1517:There are fast
1445:
1405:
1404:
1394:
1386:
1380:
1374:
1366:
1360:
1356:
1350:
1344:
1302:
1287:
1246:
1231:
1186:
1185:
1143:
1125:
1094:
1093:
1083:
1075:
1022:
1021:
973:
972:
924:
923:
880:
879:
852:
846:
799:
798:
756:
755:
718:
717:
697:
611:
610:
596:
523:
522:
416:
415:
270:
269:
226:
225:
222:
198:
181:
148:
74:
28:
23:
22:
18:Common multiple
15:
12:
11:
5:
5691:
5689:
5681:
5680:
5675:
5670:
5660:
5659:
5653:
5652:
5650:
5649:
5644:
5639:
5634:
5629:
5624:
5619:
5618:
5617:
5607:
5602:
5601:
5600:
5590:
5585:
5580:
5575:
5570:
5565:
5560:
5554:
5551:
5550:
5548:
5547:
5526:
5524:
5520:
5519:
5512:
5510:
5508:
5507:
5493:
5491:
5488:
5481:
5478:
5477:
5468:
5466:
5465:
5458:
5451:
5443:
5437:
5436:
5416:
5396:
5387:
5382:
5363:
5357:
5344:
5339:
5320:
5314:
5299:
5296:
5293:
5292:
5290:, p. 142.
5280:
5265:
5253:
5241:
5232:
5223:
5206:
5194:
5179:
5170:
5139:
5138:
5136:
5133:
5132:
5131:
5126:
5121:
5114:
5111:
4994:is an element
4960:
4957:
4956:
4955:
4944:
4941:
4936:
4932:
4928:
4925:
4920:
4917:
4914:
4910:
4906:
4903:
4900:
4895:
4891:
4887:
4882:
4878:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4848:
4844:
4840:
4837:
4834:
4829:
4825:
4821:
4816:
4812:
4808:
4805:
4802:
4791:
4790:
4776:
4772:
4768:
4765:
4762:
4757:
4753:
4749:
4744:
4740:
4720:
4719:
4708:
4703:
4700:
4697:
4694:
4690:
4686:
4682:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4647:
4643:
4640:
4637:
4634:
4631:
4628:
4624:
4609:
4608:
4583:
4580:
4579:
4578:
4567:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4414:
4400:
4399:
4388:
4385:
4382:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4301:
4298:
4288:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4164:
4163:
4162:
4161:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4122:
4118:
4115:
4112:
4102:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4060:
4056:
4053:
4050:
4040:
4033:
4030:
4029:
4028:
4017:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3983:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3945:
3935:
3923:
3922:
3921:
3920:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3857:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3794:
3792:
3785:
3782:
3781:
3780:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3690:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3588:
3581:
3578:
3577:
3576:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3516:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3463:
3460:
3450:
3437:
3436:
3428:between them:
3365:
3362:
3361:
3360:
3345:
3339:
3336:
3330:
3325:
3321:
3315:
3312:
3308:
3304:
3301:
3299:
3296:
3289:
3286:
3280:
3274:
3271:
3264:
3260:
3257:
3254:
3252:
3246:
3243:
3237:
3232:
3229:
3225:
3219:
3216:
3212:
3208:
3205:
3203:
3200:
3193:
3190:
3184:
3178:
3175:
3168:
3164:
3161:
3159:
3154:
3150:
3144:
3141:
3137:
3133:
3130:
3128:
3123:
3120:
3114:
3112:
3107:
3104:
3100:
3094:
3091:
3087:
3083:
3080:
3078:
3073:
3070:
3064:
3063:
3060:
3057:
3054:
3049:
3045:
3039:
3035:
3029:
3025:
3021:
3018:
3016:
3013:
3006:
3003:
2997:
2991:
2988:
2981:
2977:
2974:
2971:
2969:
2963:
2960:
2954:
2949:
2946:
2942:
2936:
2933:
2929:
2923:
2919:
2915:
2912:
2910:
2907:
2900:
2897:
2891:
2885:
2882:
2875:
2871:
2868:
2866:
2861:
2858:
2854:
2848:
2844:
2838:
2834:
2830:
2827:
2825:
2820:
2817:
2811:
2809:
2804:
2800:
2794:
2791:
2787:
2781:
2777:
2773:
2770:
2768:
2763:
2760:
2754:
2753:
2750:
2747:
2742:
2738:
2732:
2728:
2724:
2721:
2719:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2694:
2691:
2688:
2683:
2679:
2673:
2669:
2665:
2662:
2660:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2638:
2633:
2629:
2623:
2619:
2615:
2612:
2610:
2608:
2605:
2603:
2598:
2594:
2588:
2584:
2580:
2577:
2575:
2573:
2570:
2569:
2555:
2554:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2481:
2480:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2404:
2403:
2392:
2387:
2382:
2378:
2374:
2369:
2365:
2361:
2358:
2354:
2348:
2344:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2305:
2304:
2291:
2286:
2282:
2278:
2273:
2269:
2265:
2262:
2258:
2252:
2248:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2199:
2195:
2190:
2184:
2180:
2176:
2173:
2149:
2145:
2140:
2134:
2130:
2126:
2123:
2108:
2101:
2095:
2094:
2083:
2076:
2072:
2067:
2061:
2057:
2053:
2050:
2043:
2039:
2034:
2026:
2022:
2017:
2009:
2005:
2000:
1992:
1988:
1983:
1979:
1976:
1953:
1950:
1948:
1945:
1937:
1936:
1933:
1930:
1926:
1925:
1914:
1913:
1910:
1881:
1880:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1843:
1839:
1835:
1830:
1826:
1822:
1817:
1813:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1768:
1767:
1750:
1746:
1742:
1737:
1733:
1729:
1726:
1724:
1722:
1719:
1718:
1713:
1709:
1705:
1702:
1700:
1698:
1695:
1694:
1689:
1685:
1681:
1678:
1676:
1674:
1671:
1670:
1643:
1642:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1605:
1601:
1597:
1592:
1588:
1584:
1579:
1575:
1571:
1568:
1538:
1535:
1521:, such as the
1515:
1514:
1503:
1500:
1497:
1494:
1491:
1488:
1483:
1480:
1475:
1472:
1469:
1463:
1460:
1457:
1454:
1451:
1448:
1444:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1338:
1337:
1326:
1320:
1317:
1314:
1311:
1308:
1305:
1299:
1295:
1291:
1282:
1278:
1274:
1270:
1264:
1261:
1258:
1255:
1252:
1249:
1243:
1239:
1235:
1226:
1222:
1218:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1179:
1178:
1167:
1161:
1158:
1155:
1152:
1149:
1146:
1140:
1136:
1133:
1129:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1082:
1079:
1074:
1071:
1058:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1009:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
960:
956:
953:
950:
947:
944:
941:
938:
935:
932:
911:
908:
905:
902:
899:
896:
893:
890:
887:
845:
842:
829:
825:
822:
819:
816:
813:
810:
807:
786:
782:
779:
776:
773:
770:
767:
764:
743:
740:
737:
734:
731:
728:
725:
696:
693:
689:
688:
675:
672:
667:
662:
659:
654:
649:
646:
641:
636:
633:
628:
623:
620:
595:
592:
584:
583:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
513:
512:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
409:
408:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
286:
283:
280:
277:
263:
262:
251:
248:
245:
242:
239:
236:
233:
221:
218:
197:
194:
180:
177:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5690:
5679:
5678:Number theory
5676:
5674:
5671:
5669:
5666:
5665:
5663:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5616:
5613:
5612:
5611:
5608:
5606:
5603:
5599:
5596:
5595:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5561:
5559:
5556:
5555:
5552:
5528:
5527:
5525:
5521:
5506:
5502:
5498:
5495:
5494:
5492:
5485:
5479:
5475:
5471:
5464:
5459:
5457:
5452:
5450:
5445:
5444:
5441:
5434:
5430:
5426:
5425:Prentice Hall
5422:
5417:
5414:
5410:
5406:
5402:
5397:
5393:
5388:
5385:
5379:
5375:
5371:
5370:
5364:
5360:
5354:
5350:
5345:
5342:
5340:0-387-94777-9
5336:
5332:
5328:
5327:
5321:
5317:
5311:
5307:
5302:
5301:
5297:
5289:
5284:
5281:
5278:, p. 94.
5277:
5272:
5270:
5266:
5263:, p. 58)
5262:
5257:
5254:
5251:, p. 41)
5250:
5245:
5242:
5236:
5233:
5227:
5224:
5216:
5210:
5207:
5204:, p. 56)
5203:
5198:
5195:
5192:, p. 39)
5191:
5186:
5184:
5180:
5174:
5171:
5160:
5156:
5149:
5147:
5145:
5141:
5134:
5130:
5127:
5125:
5122:
5120:
5117:
5116:
5112:
5110:
5092:
5088:
5084:
5079:
5074:divides
5049:
5044:
5040:
5034:
5030:
4985:
4968:
4966:
4958:
4942:
4934:
4930:
4926:
4918:
4915:
4912:
4908:
4904:
4901:
4898:
4893:
4889:
4885:
4880:
4876:
4869:
4866:
4860:
4857:
4854:
4846:
4842:
4838:
4835:
4832:
4827:
4823:
4819:
4814:
4810:
4803:
4800:
4793:
4792:
4789:is zero, then
4774:
4770:
4766:
4763:
4760:
4755:
4751:
4747:
4742:
4738:
4729:
4728:
4727:
4725:
4706:
4698:
4692:
4688:
4684:
4673:
4670:
4664:
4661:
4658:
4652:
4649:
4645:
4638:
4635:
4632:
4614:
4613:
4612:
4606:
4602:
4598:
4594:
4590:
4586:
4585:
4581:
4565:
4556:
4553:
4550:
4541:
4535:
4532:
4529:
4520:
4514:
4511:
4508:
4496:
4493:
4490:
4481:
4478:
4475:
4469:
4466:
4463:
4457:
4454:
4451:
4445:
4442:
4439:
4433:
4430:
4427:
4421:
4418:
4405:
4404:
4403:
4386:
4377:
4374:
4371:
4362:
4356:
4353:
4350:
4338:
4335:
4332:
4323:
4320:
4317:
4311:
4308:
4305:
4302:
4289:
4275:
4266:
4263:
4260:
4254:
4251:
4248:
4242:
4239:
4236:
4230:
4227:
4218:
4209:
4206:
4203:
4194:
4191:
4185:
4182:
4175:
4174:
4173:
4171:
4148:
4142:
4139:
4136:
4127:
4124:
4116:
4113:
4110:
4103:
4089:
4083:
4080:
4077:
4071:
4068:
4065:
4062:
4054:
4051:
4048:
4041:
4039:
4036:
4035:
4034:
4031:
4015:
4012:
4009:
4003:
4000:
3997:
3984:
3970:
3967:
3964:
3958:
3955:
3952:
3946:
3943:
3936:
3934:
3931:
3930:
3929:
3928:
3907:
3904:
3901:
3892:
3889:
3886:
3880:
3877:
3874:
3871:
3858:
3844:
3841:
3838:
3829:
3826:
3823:
3814:
3811:
3805:
3802:
3795:
3793:
3791:
3788:
3787:
3786:
3783:
3767:
3761:
3758:
3752:
3749:
3746:
3731:
3722:
3719:
3716:
3707:
3704:
3691:
3677:
3671:
3668:
3662:
3659:
3656:
3650:
3647:
3641:
3638:
3635:
3626:
3623:
3620:
3614:
3611:
3608:
3605:
3599:
3596:
3589:
3587:
3584:
3583:
3582:
3579:
3563:
3557:
3554:
3551:
3542:
3536:
3533:
3530:
3517:
3503:
3497:
3494:
3491:
3485:
3482:
3479:
3473:
3470:
3467:
3461:
3458:
3451:
3449:
3446:
3445:
3444:
3443:
3440:
3434:
3431:
3430:
3429:
3427:
3422:
3418:
3414:
3409:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3380:(that is, if
3379:
3375:
3371:
3363:
3343:
3337:
3334:
3328:
3323:
3319:
3313:
3310:
3306:
3302:
3300:
3294:
3287:
3284:
3278:
3272:
3269:
3262:
3258:
3255:
3250:
3244:
3241:
3235:
3230:
3227:
3223:
3217:
3214:
3210:
3206:
3204:
3198:
3191:
3188:
3182:
3176:
3173:
3166:
3157:
3152:
3148:
3142:
3139:
3135:
3131:
3129:
3121:
3118:
3110:
3105:
3102:
3098:
3092:
3089:
3085:
3081:
3079:
3071:
3068:
3058:
3055:
3052:
3047:
3043:
3037:
3033:
3027:
3023:
3019:
3017:
3011:
3004:
3001:
2995:
2989:
2986:
2979:
2975:
2972:
2967:
2961:
2958:
2952:
2947:
2944:
2940:
2934:
2931:
2927:
2921:
2917:
2913:
2911:
2905:
2898:
2895:
2889:
2883:
2880:
2873:
2864:
2859:
2856:
2852:
2846:
2842:
2836:
2832:
2828:
2826:
2818:
2815:
2807:
2802:
2798:
2792:
2789:
2785:
2779:
2775:
2771:
2769:
2761:
2758:
2748:
2745:
2740:
2736:
2730:
2726:
2722:
2720:
2712:
2709:
2706:
2700:
2697:
2692:
2689:
2686:
2681:
2677:
2671:
2667:
2663:
2661:
2653:
2650:
2647:
2636:
2631:
2627:
2621:
2617:
2613:
2611:
2606:
2601:
2596:
2592:
2586:
2582:
2578:
2576:
2571:
2560:
2559:
2558:
2541:
2538:
2535:
2532:
2526:
2523:
2520:
2514:
2511:
2505:
2502:
2499:
2486:
2485:
2484:
2467:
2464:
2461:
2458:
2455:
2449:
2446:
2443:
2434:
2428:
2425:
2422:
2409:
2408:
2407:
2390:
2380:
2376:
2372:
2367:
2363:
2352:
2346:
2342:
2338:
2332:
2329:
2326:
2320:
2317:
2310:
2309:
2308:
2284:
2280:
2276:
2271:
2267:
2256:
2250:
2246:
2242:
2236:
2233:
2230:
2217:
2216:
2215:
2197:
2193:
2188:
2182:
2178:
2174:
2171:
2147:
2143:
2138:
2132:
2128:
2124:
2121:
2112:
2107:
2100:
2081:
2074:
2070:
2065:
2059:
2055:
2051:
2048:
2041:
2037:
2032:
2024:
2020:
2015:
2007:
2003:
1998:
1990:
1986:
1981:
1977:
1974:
1967:
1966:
1965:
1963:
1959:
1951:
1946:
1944:
1942:
1934:
1931:
1928:
1927:
1923:
1919:
1918:
1917:
1911:
1908:
1907:
1906:
1903:
1901:
1897:
1893:
1888:
1886:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1841:
1837:
1833:
1828:
1824:
1820:
1815:
1811:
1807:
1801:
1798:
1795:
1792:
1789:
1783:
1780:
1773:
1772:
1771:
1748:
1744:
1740:
1735:
1731:
1727:
1725:
1720:
1711:
1707:
1703:
1701:
1696:
1687:
1683:
1679:
1677:
1672:
1661:
1660:
1659:
1657:
1652:
1649:
1646:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1603:
1599:
1595:
1590:
1586:
1582:
1577:
1573:
1569:
1566:
1559:
1558:
1557:
1556:For example:
1554:
1552:
1548:
1547:prime numbers
1544:
1536:
1534:
1532:
1528:
1524:
1520:
1501:
1498:
1495:
1492:
1489:
1486:
1481:
1478:
1473:
1470:
1467:
1458:
1455:
1452:
1442:
1437:
1434:
1431:
1425:
1422:
1419:
1413:
1410:
1403:
1402:
1401:
1398:
1395:lcm(0, 0) = 0
1392:
1383:
1377:
1369:
1365:, 0) = |
1364:
1353:
1347:
1341:
1324:
1315:
1312:
1309:
1293:
1276:
1268:
1259:
1256:
1253:
1237:
1220:
1212:
1206:
1203:
1200:
1194:
1191:
1184:
1183:
1182:
1165:
1156:
1153:
1150:
1134:
1131:
1120:
1114:
1111:
1108:
1102:
1099:
1092:
1091:
1090:
1088:
1080:
1078:
1072:
1070:
1056:
1049:
1046:
1043:
1040:
1037:
1031:
1028:
1007:
1000:
997:
994:
991:
988:
982:
979:
958:
951:
948:
945:
942:
939:
933:
930:
906:
903:
900:
897:
894:
888:
885:
877:
873:
869:
865:
861:
857:
851:
843:
841:
827:
820:
817:
814:
808:
805:
784:
777:
774:
771:
765:
762:
738:
735:
732:
726:
723:
714:
710:
706:
702:
701:meshing gears
695:Gears problem
694:
692:
673:
670:
665:
660:
657:
652:
647:
644:
639:
634:
631:
626:
621:
618:
609:
608:
607:
605:
601:
593:
591:
589:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
521:
520:
519:
517:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
421:
414:
413:
412:
395:
392:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
268:
267:
266:
246:
243:
240:
234:
231:
224:
223:
219:
217:
215:
211:
207:
203:
195:
193:
190:
186:
178:
176:
174:
170:
166:
160:
156:
152:
146:
142:
138:
133:
131:
127:
122:
120:
116:
112:
108:
104:
100:
96:
92:
88:
82:
78:
72:
68:
65:
61:
57:
53:
49:
48:number theory
45:
37:
32:
19:
5420:
5400:
5391:
5368:
5348:
5329:, New York:
5325:
5305:
5288:Grillet 2007
5283:
5256:
5244:
5235:
5226:
5209:
5197:
5173:
5162:. Retrieved
5158:
5080:
5047:
5042:
5038:
5032:
5028:
4983:
4969:
4967:as follows:
4962:
4721:
4610:
4600:
4596:
4592:
4588:
4401:
4170:distributive
4167:
3438:
3432:
3410:
3405:
3401:
3397:
3393:
3389:
3381:
3377:
3373:
3367:
2556:
2482:
2405:
2306:
2113:
2105:
2098:
2096:
1955:
1938:
1915:
1904:
1899:
1892:Venn diagram
1889:
1882:
1769:
1653:
1650:
1647:
1644:
1555:
1540:
1516:
1399:
1381:
1375:
1367:
1362:
1351:
1345:
1342:
1339:
1180:
1084:
1076:
875:
871:
867:
863:
859:
855:
853:
712:
708:
698:
690:
597:
594:Applications
590:is 12.
587:
585:
515:
514:
410:
264:
213:
209:
205:
201:
199:
182:
172:
168:
164:
158:
154:
150:
144:
140:
136:
134:
123:
118:
114:
110:
106:
102:
94:
90:
80:
76:
70:
66:
59:
55:
51:
41:
36:Venn diagram
5610:Irreducible
5540:Denominator
5276:Burton 1970
4730:If none of
4724:cardinality
2483:this gives
1073:Calculation
840:rotations.
5662:Categories
5642:Percentage
5637:Paper size
5546:= Quotient
5372:, Oxford:
5298:References
5249:Long (1972
5190:Long (1972
5164:2020-08-30
5083:associates
5026:such that
4726:of a set.
4605:squarefree
1519:algorithms
848:See also:
44:arithmetic
5615:Reduction
5573:Continued
5558:Algebraic
5534:Numerator
5470:Fractions
5413:77-171950
4916:−
4902:…
4870:
4861:
4836:…
4804:
4764:…
4693:ω
4653:
4497:
4470:
4446:
4422:
4339:
4312:
4255:
4231:
4186:
4121:⟺
4114:≤
4072:
4059:⟺
4052:≥
3947:
3881:
3806:
3651:
3642:
3615:
3600:
3486:
3462:
3311:−
3259:
3228:−
3215:−
3140:−
3103:−
3090:−
2976:
2945:−
2932:−
2857:−
2790:−
2701:
2515:
2343:∏
2321:
2247:∏
2179:∏
2129:∏
2056:∏
2049:⋯
1859:⋅
1853:⋅
1834:⋅
1821:⋅
1784:
1741:⋅
1627:⋅
1621:⋅
1615:⋅
1596:⋅
1583:⋅
1493:×
1474:×
1438:×
1414:
1195:
1103:
1032:
983:
934:
889:
809:
766:
727:
707:, having
235:
126:fractions
97:. Since
87:divisible
5588:Egyptian
5523:Fraction
5505:Quotient
5497:Dividend
5433:77-81766
5331:Springer
5113:See also
3392:) write
3376:divides
1947:Formulas
1527:factored
1359:, since
196:Notation
185:multiple
179:Overview
175:, . . .
161:, . . .)
128:is the "
89:by both
64:integers
5605:Integer
5578:Decimal
5543:
5531:
5501:Divisor
5085:. In a
5010:divide
3426:duality
3415:, with
3413:lattice
705:machine
220:Example
189:product
157:,
153:,
121:and 0.
79:,
62:of two
5598:Silver
5593:Golden
5583:Dyadic
5568:Binary
5563:Aspect
5474:ratios
5431:
5411:
5380:
5355:
5337:
5312:
3384:is an
2406:Since
1656:powers
1393:; so,
1371:|
50:, the
5218:(PDF)
5135:Notes
5046:). A
4611:Then
4582:Other
1962:up to
703:in a
58:, or
5647:Unit
5472:and
5429:LCCN
5409:LCCN
5378:ISBN
5353:ISBN
5335:ISBN
5310:ISBN
5105:and
5097:and
5066:and
5054:and
5036:and
5018:and
5006:and
4990:and
4982:. A
4974:and
4970:Let
4587:Let
3421:join
3417:meet
2307:and
2164:and
1868:504.
1541:The
1385:are
1379:and
1361:gcd(
1349:and
1020:and
874:and
862:and
711:and
204:and
149:lcm(
105:and
93:and
75:lcm(
69:and
46:and
5627:LCD
5062:of
5050:of
5022:of
4998:of
4986:of
4867:lcm
4858:lcm
4801:lcm
4650:lcm
4603:is
4545:gcd
4524:gcd
4503:gcd
4494:lcm
4467:lcm
4443:lcm
4419:lcm
4413:gcd
4366:gcd
4345:gcd
4336:lcm
4309:lcm
4297:gcd
4252:lcm
4228:lcm
4222:gcd
4198:gcd
4183:lcm
4131:gcd
4069:lcm
3992:gcd
3944:lcm
3878:lcm
3866:gcd
3818:gcd
3803:lcm
3741:gcd
3735:gcd
3711:gcd
3699:gcd
3648:lcm
3639:lcm
3612:lcm
3597:lcm
3546:gcd
3525:gcd
3483:lcm
3459:lcm
3396:≤
3388:of
3256:lcm
3163:gcd
2973:lcm
2870:gcd
2749:12.
2698:lcm
2642:gcd
2512:lcm
2494:gcd
2438:max
2417:min
2357:max
2318:lcm
2261:min
2225:gcd
1900:all
1781:lcm
1502:42.
1447:gcd
1411:lcm
1355:is
1304:gcd
1248:gcd
1192:lcm
1145:gcd
1100:lcm
1029:lcm
980:lcm
931:lcm
886:lcm
806:lcm
763:lcm
724:lcm
232:lcm
42:In
5664::
5503:=
5499:÷
5427:,
5407:,
5376:,
5333:,
5268:^
5182:^
5157:.
5143:^
5078:.
5070:,
5041:=
5039:by
5031:=
5029:ax
4607:).
3404:≥
3245:12
2962:15
2104:,
1887:.
1802:21
1721:21
1658:.
1630:5.
1567:90
1553:.
1479:21
1453:21
1443:21
1420:21
971:,
870:,
858:,
674:42
671:11
661:42
648:42
622:21
559:72
553:60
547:48
541:36
535:24
529:12
488:72
482:66
476:60
470:54
464:48
458:42
452:36
446:30
440:24
434:18
428:12
384:76
378:72
372:68
366:64
360:60
354:56
348:52
342:48
336:44
330:40
324:36
318:32
312:28
306:24
300:20
294:16
288:12
212:,
183:A
171:,
167:,
143:,
139:,
54:,
34:A
5537:/
5462:e
5455:t
5448:v
5361:.
5318:.
5220:.
5167:.
5107:b
5103:a
5099:b
5095:a
5076:n
5072:m
5068:b
5064:a
5060:n
5056:b
5052:a
5043:m
5033:m
5024:R
5020:y
5016:x
5012:m
5008:b
5004:a
5000:R
4996:m
4992:b
4988:a
4980:R
4976:b
4972:a
4943:.
4940:)
4935:r
4931:a
4927:,
4924:)
4919:1
4913:r
4909:a
4905:,
4899:,
4894:2
4890:a
4886:,
4881:1
4877:a
4873:(
4864:(
4855:=
4852:)
4847:r
4843:a
4839:,
4833:,
4828:2
4824:a
4820:,
4815:1
4811:a
4807:(
4775:r
4771:a
4767:,
4761:,
4756:2
4752:a
4748:,
4743:1
4739:a
4707:,
4702:)
4699:D
4696:(
4689:3
4685:=
4681:|
4677:}
4674:D
4671:=
4668:)
4665:y
4662:,
4659:x
4656:(
4646::
4642:)
4639:y
4636:,
4633:x
4630:(
4627:{
4623:|
4601:D
4597:D
4595:(
4593:ω
4589:D
4566:.
4563:)
4560:)
4557:c
4554:,
4551:a
4548:(
4542:,
4539:)
4536:c
4533:,
4530:b
4527:(
4521:,
4518:)
4515:b
4512:,
4509:a
4506:(
4500:(
4491:=
4488:)
4485:)
4482:c
4479:,
4476:a
4473:(
4464:,
4461:)
4458:c
4455:,
4452:b
4449:(
4440:,
4437:)
4434:b
4431:,
4428:a
4425:(
4416:(
4387:.
4384:)
4381:)
4378:c
4375:,
4372:a
4369:(
4363:,
4360:)
4357:b
4354:,
4351:a
4348:(
4342:(
4333:=
4330:)
4327:)
4324:c
4321:,
4318:b
4315:(
4306:,
4303:a
4300:(
4276:,
4273:)
4270:)
4267:c
4264:,
4261:a
4258:(
4249:,
4246:)
4243:b
4240:,
4237:a
4234:(
4225:(
4219:=
4216:)
4213:)
4210:c
4207:,
4204:b
4201:(
4195:,
4192:a
4189:(
4149:.
4146:)
4143:b
4140:,
4137:a
4134:(
4128:=
4125:a
4117:b
4111:a
4090:,
4087:)
4084:b
4081:,
4078:a
4075:(
4066:=
4063:a
4055:b
4049:a
4016:.
4013:a
4010:=
4007:)
4004:a
4001:,
3998:a
3995:(
3971:,
3968:a
3965:=
3962:)
3959:a
3956:,
3953:a
3950:(
3908:.
3905:a
3902:=
3899:)
3896:)
3893:b
3890:,
3887:a
3884:(
3875:,
3872:a
3869:(
3845:,
3842:a
3839:=
3836:)
3833:)
3830:b
3827:,
3824:a
3821:(
3815:,
3812:a
3809:(
3768:.
3765:)
3762:c
3759:,
3756:)
3753:b
3750:,
3747:a
3744:(
3738:(
3732:=
3729:)
3726:)
3723:c
3720:,
3717:b
3714:(
3708:,
3705:a
3702:(
3678:,
3675:)
3672:c
3669:,
3666:)
3663:b
3660:,
3657:a
3654:(
3645:(
3636:=
3633:)
3630:)
3627:c
3624:,
3621:b
3618:(
3609:,
3606:a
3603:(
3564:.
3561:)
3558:a
3555:,
3552:b
3549:(
3543:=
3540:)
3537:b
3534:,
3531:a
3528:(
3504:,
3501:)
3498:a
3495:,
3492:b
3489:(
3480:=
3477:)
3474:b
3471:,
3468:a
3465:(
3406:a
3402:b
3398:b
3394:a
3390:a
3382:b
3378:b
3374:a
3344:.
3338:2
3335:3
3329:=
3324:1
3320:3
3314:1
3307:2
3303:=
3295:)
3288:4
3285:3
3279:,
3273:6
3270:1
3263:(
3251:,
3242:1
3236:=
3231:1
3224:3
3218:2
3211:2
3207:=
3199:)
3192:4
3189:3
3183:,
3177:6
3174:1
3167:(
3158:,
3153:1
3149:3
3143:2
3136:2
3132:=
3122:4
3119:3
3111:,
3106:1
3099:3
3093:1
3086:2
3082:=
3072:6
3069:1
3059:,
3056:2
3053:=
3048:0
3044:5
3038:0
3034:3
3028:1
3024:2
3020:=
3012:)
3005:5
3002:2
2996:,
2990:3
2987:1
2980:(
2968:,
2959:1
2953:=
2948:1
2941:5
2935:1
2928:3
2922:0
2918:2
2914:=
2906:)
2899:5
2896:2
2890:,
2884:3
2881:1
2874:(
2865:,
2860:1
2853:5
2847:0
2843:3
2837:1
2833:2
2829:=
2819:5
2816:2
2808:,
2803:0
2799:5
2793:1
2786:3
2780:0
2776:2
2772:=
2762:3
2759:1
2746:=
2741:1
2737:3
2731:2
2727:2
2723:=
2716:)
2713:6
2710:,
2707:4
2704:(
2693:,
2690:2
2687:=
2682:0
2678:3
2672:1
2668:2
2664:=
2657:)
2654:6
2651:,
2648:4
2645:(
2637:,
2632:1
2628:3
2622:1
2618:2
2614:=
2607:6
2602:,
2597:0
2593:3
2587:2
2583:2
2579:=
2572:4
2542:.
2539:b
2536:a
2533:=
2530:)
2527:b
2524:,
2521:a
2518:(
2509:)
2506:b
2503:,
2500:a
2497:(
2468:,
2465:y
2462:+
2459:x
2456:=
2453:)
2450:y
2447:,
2444:x
2441:(
2435:+
2432:)
2429:y
2426:,
2423:x
2420:(
2391:.
2386:)
2381:p
2377:b
2373:,
2368:p
2364:a
2360:(
2353:p
2347:p
2339:=
2336:)
2333:b
2330:,
2327:a
2324:(
2290:)
2285:p
2281:b
2277:,
2272:p
2268:a
2264:(
2257:p
2251:p
2243:=
2240:)
2237:b
2234:,
2231:a
2228:(
2198:p
2194:b
2189:p
2183:p
2175:=
2172:b
2148:p
2144:a
2139:p
2133:p
2125:=
2122:a
2109:3
2106:n
2102:2
2099:n
2082:,
2075:p
2071:n
2066:p
2060:p
2052:=
2042:7
2038:n
2033:7
2025:5
2021:n
2016:5
2008:3
2004:n
1999:3
1991:2
1987:n
1982:2
1978:=
1975:n
1865:=
1862:7
1856:9
1850:8
1847:=
1842:1
1838:7
1829:2
1825:3
1816:3
1812:2
1808:=
1805:)
1799:,
1796:9
1793:,
1790:8
1787:(
1749:1
1745:7
1736:1
1732:3
1728:=
1712:2
1708:3
1704:=
1697:9
1688:3
1684:2
1680:=
1673:8
1624:3
1618:3
1612:2
1609:=
1604:1
1600:5
1591:2
1587:3
1578:1
1574:2
1570:=
1499:=
1496:7
1490:6
1487:=
1482:3
1471:6
1468:=
1462:)
1459:6
1456:,
1450:(
1435:6
1432:=
1429:)
1426:6
1423:,
1417:(
1387:0
1382:b
1376:a
1368:a
1363:a
1357:0
1352:b
1346:a
1325:,
1319:)
1316:b
1313:,
1310:a
1307:(
1298:|
1294:a
1290:|
1281:|
1277:b
1273:|
1269:=
1263:)
1260:b
1257:,
1254:a
1251:(
1242:|
1238:b
1234:|
1225:|
1221:a
1217:|
1213:=
1210:)
1207:b
1204:,
1201:a
1198:(
1166:.
1160:)
1157:b
1154:,
1151:a
1148:(
1139:|
1135:b
1132:a
1128:|
1121:=
1118:)
1115:b
1112:,
1109:a
1106:(
1057:n
1053:)
1050:n
1047:,
1044:m
1041:,
1038:l
1035:(
1008:m
1004:)
1001:n
998:,
995:m
992:,
989:l
986:(
959:l
955:)
952:n
949:,
946:m
943:,
940:l
937:(
910:)
907:n
904:,
901:m
898:,
895:l
892:(
876:n
872:m
868:l
864:n
860:m
856:l
828:n
824:)
821:n
818:,
815:m
812:(
785:m
781:)
778:n
775:,
772:m
769:(
742:)
739:n
736:,
733:m
730:(
713:n
709:m
666:=
658:7
653:+
645:4
640:=
635:6
632:1
627:+
619:2
571:.
568:.
565:.
562:,
556:,
550:,
544:,
538:,
532:,
500:.
497:.
494:.
491:,
485:,
479:,
473:,
467:,
461:,
455:,
449:,
443:,
437:,
431:,
425:,
422:6
396:.
393:.
390:.
387:,
381:,
375:,
369:,
363:,
357:,
351:,
345:,
339:,
333:,
327:,
321:,
315:,
309:,
303:,
297:,
291:,
285:,
282:8
279:,
276:4
250:)
247:6
244:,
241:4
238:(
214:b
210:a
206:b
202:a
173:c
169:b
165:a
159:c
155:b
151:a
145:c
141:b
137:a
119:a
115:a
111:a
107:b
103:a
95:b
91:a
83:)
81:b
77:a
71:b
67:a
20:)
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