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Statement | ||
According to Wikipedia ("Least common multiple", 27-Aug-2020, https://en.wikipedia.org/wiki/Least_common_multiple): "In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the lcm to be the least upper bound in the lattice of divisibility. ... The lcm of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them." In this section, an operation calculating the least common multiple of two integers (df-lcm 15303) as well as a function mapping a set of integers to their least common multiple (df-lcmf 15304) are provided. Both definitions are valid for all integers, including negative integers and 0, obeying the above mentioned convention. It is shown by lcmfpr 15340 that the two definitions are compatible. | ||
Syntax | clcm 15301 | Extend the definition of a class to include the least common multiple operator. |
lcm | ||
Syntax | clcmf 15302 | Extend the definition of a class to include the least common multiple function. |
lcm | ||
Definition | df-lcm 15303* | Define the lcm operator. For example, lcm (ex-lcm 27315). (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Definition | df-lcmf 15304* | Define the lcm function on a set of integers. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmval 15305* | Value of the lcm operator. lcm is the least common multiple of and . If either or is , the result is defined conventionally as . Contrast with df-gcd 15217 and gcdval 15218. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmcom 15306 | The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcm0val 15307 | The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 15306 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmn0val 15308* | The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmcllem 15309* | Lemma for lcmn0cl 15310 and dvdslcm 15311. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmn0cl 15310 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | dvdslcm 15311 | The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmledvds 15312 | A positive integer which both operands of the lcm operator divide bounds it. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmeq0 15313 | The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmcl 15314 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | gcddvdslcm 15315 | The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmneg 15316 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | neglcm 15317 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmabs 15318 | The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmgcdlem 15319 | Lemma for lcmgcd 15320 and lcmdvds 15321. Prove them for positive , , and . (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcmgcd 15320 |
The product of two numbers' least common multiple and greatest common
divisor is the absolute value of the product of the two numbers. In
particular, that absolute value is the least common multiple of two
coprime numbers, for which .
Multiple methods exist for proving this, and it is often proven either as a consequence of the fundamental theorem of arithmetic 1arith 15631 or of Bézout's identity bezout 15260; see e.g. https://proofwiki.org/wiki/Product_of_GCD_and_LCM and https://math.stackexchange.com/a/470827. This proof uses the latter to first confirm it for positive integers and (the "Second Proof" in the above Stack Exchange page), then shows that implies it for all nonzero integer inputs, then finally uses lcm0val 15307 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmdvds 15321 | The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmid 15322 | The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcm1 15323 | The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.) |
lcm | ||
Theorem | lcmgcdnn 15324 | The product of two positive integers' least common multiple and greatest common divisor is the product of the two integers. (Contributed by AV, 27-Aug-2020.) |
lcm | ||
Theorem | lcmgcdeq 15325 | Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmdvdsb 15326 | Biconditional form of lcmdvds 15321. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmass 15327 | Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm lcm lcm | ||
Theorem | 3lcm2e6woprm 15328 | The least common multiple of three and two is six. In contrast to 3lcm2e6 15440, this proof does not use the property of 2 and 3 being prime, therefore it is much longer. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
lcm | ||
Theorem | 6lcm4e12 15329 | The least common multiple of six and four is twelve. (Contributed by AV, 27-Aug-2020.) |
lcm ; | ||
Theorem | absproddvds 15330* | The absolute value of the product of the elements of a finite subset of the integers is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.) |
Theorem | absprodnn 15331* | The absolute value of the product of the elements of a finite subset of the integers not containing 0 is a poitive integer. (Contributed by AV, 21-Aug-2020.) |
Theorem | fissn0dvds 15332* | For each finite subset of the integers not containing 0 there is a positive integer which is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.) |
Theorem | fissn0dvdsn0 15333* | For each finite subset of the integers not containing 0 there is a positive integer which is divisible by each element of this subset. (Contributed by AV, 21-Aug-2020.) |
Theorem | lcmfval 15334* | Value of the lcm function. lcm is the least common multiple of the integers contained in the finite subset of integers . If at least one of the elements of is , the result is defined conventionally as . (Contributed by AV, 21-Apr-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmf0val 15335 | The value, by convention, of the least common multiple for a set containing 0 is 0. (Contributed by AV, 21-Apr-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmfn0val 15336* | The value of the lcm function for a set without 0. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmfnnval 15337* | The value of the lcm function for a subset of the positive integers. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmfcllem 15338* | Lemma for lcmfn0cl 15339 and dvdslcmf 15344. (Contributed by AV, 21-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmfn0cl 15339 | Closure of the lcm function. (Contributed by AV, 21-Aug-2020.) |
lcm | ||
Theorem | lcmfpr 15340 | The value of the lcm function for an unordered pair is the value of the lcm operator for both elements. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcmfcl 15341 | Closure of the lcm function. (Contributed by AV, 21-Aug-2020.) |
lcm | ||
Theorem | lcmfnncl 15342 | Closure of the lcm function. (Contributed by AV, 20-Apr-2020.) |
lcm | ||
Theorem | lcmfeq0b 15343 | The least common multiple of a set of integers is 0 iff at least one of its element is 0. (Contributed by AV, 21-Aug-2020.) |
lcm | ||
Theorem | dvdslcmf 15344* | The least common multiple of a set of integers is divisible by each of its elements. (Contributed by AV, 22-Aug-2020.) |
lcm | ||
Theorem | lcmfledvds 15345* | A positive integer which is divisible by all elements of a set of integers bounds the least common multiple of the set. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmf 15346* | Characterization of the least common multiple of a set of integers (without 0): A positiven integer is the least common multiple of a set of integers iff it divides each of the elements of the set and every integer which divides each of the elements of the set is greater than or equal to this integer. (Contributed by AV, 22-Aug-2020.) |
lcm | ||
Theorem | lcmf0 15347 | The least common multiple of the empty set is 1. (Contributed by AV, 22-Aug-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmfsn 15348 | The least common multiple of a singleton is its absolute value. (Contributed by AV, 22-Aug-2020.) |
lcm | ||
Theorem | lcmftp 15349 | The least common multiple of a triple of integers is the least common multiple of the third integer and the the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 15357, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.) |
lcm lcm lcm | ||
Theorem | lcmfunsnlem1 15350* | Lemma for lcmfdvds 15355 and lcmfunsnlem 15354 (Induction step part 1). (Contributed by AV, 25-Aug-2020.) |
lcm lcm lcm lcm lcm | ||
Theorem | lcmfunsnlem2lem1 15351* | Lemma 1 for lcmfunsnlem2 15353. (Contributed by AV, 26-Aug-2020.) |
lcm lcm lcm lcm lcm lcm | ||
Theorem | lcmfunsnlem2lem2 15352* | Lemma 2 for lcmfunsnlem2 15353. (Contributed by AV, 26-Aug-2020.) |
lcm lcm lcm lcm lcm lcm lcm | ||
Theorem | lcmfunsnlem2 15353* | Lemma for lcmfunsn 15357 and lcmfunsnlem 15354 (Induction step part 2). (Contributed by AV, 26-Aug-2020.) |
lcm lcm lcm lcm lcm lcm lcm | ||
Theorem | lcmfunsnlem 15354* | Lemma for lcmfdvds 15355 and lcmfunsn 15357. These two theorems must be proven simultaneously by induction on the cardinality of a finite set , because they depend on each other. This can be seen by the two parts lcmfunsnlem1 15350 and lcmfunsnlem2 15353 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.) |
lcm lcm lcm lcm | ||
Theorem | lcmfdvds 15355* | The least common multiple of a set of integers divides any integer which is divisible by all elements of the set. (Contributed by AV, 26-Aug-2020.) |
lcm | ||
Theorem | lcmfdvdsb 15356* | Biconditional form of lcmfdvds 15355. (Contributed by AV, 26-Aug-2020.) |
lcm | ||
Theorem | lcmfunsn 15357 | The lcm function for a union of a set of integer and a singleton. (Contributed by AV, 26-Aug-2020.) |
lcm lcm lcm | ||
Theorem | lcmfun 15358 | The lcm function for a union of sets of integers. (Contributed by AV, 27-Aug-2020.) |
lcm lcm lcm lcm | ||
Theorem | lcmfass 15359 | Associative law for the lcm function. (Contributed by AV, 27-Aug-2020.) |
lcmlcm lcm lcm | ||
Theorem | lcmf2a3a4e12 15360 | The least common multiple of 2 , 3 and 4 is 12. (Contributed by AV, 27-Aug-2020.) |
lcm ; | ||
Theorem | lcmflefac 15361 | The least common multiple of all positive integers less than or equal to an integer is less than or equal to the factorial of the integer. (Contributed by AV, 16-Aug-2020.) (Revised by AV, 27-Aug-2020.) |
lcm | ||
According to Wikipedia "Coprime integers", see https://en.wikipedia.org/wiki/Coprime_integers (16-Aug-2020) "[...] two integers a and b are said to be relatively prime, mutually prime, or coprime [...] if the only positive integer (factor) that divides both of them is 1. Consequently, any prime number that divides one does not divide the other. This is equivalent to their greatest common divisor (gcd) being 1.". In the following, we use this equivalent characterization to say that and are coprime (or relatively prime) if . The equivalence of the definitions is shown by coprmgcdb 15362. The negation, i.e. two integers are not coprime, can be expressed either by , see ncoprmgcdne1b 15363, or equivalently by , see ncoprmgcdgt1b 15364. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 15366 (see euclemma 15425 for a version of Euclid's lemma for primes). | ||
Theorem | coprmgcdb 15362* | Two positive integers are coprime, i.e. the only positive integer that divides both of them is 1, iff their greatest common divisor is 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | ncoprmgcdne1b 15363* | Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is not 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | ncoprmgcdgt1b 15364* | Two positive integers are not coprime, i.e. there is an integer greater than 1 which divides both integers, iff their greatest common divisor is greater than 1. (Contributed by AV, 9-Aug-2020.) |
Theorem | coprmdvds1 15365 | If two positive integers are coprime, i.e. their greatest common divisor is 1, the only positive integer that divides both of them is 1. (Contributed by AV, 4-Aug-2021.) |
Theorem | coprmdvds 15366 | Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. Generalization of euclemma 15425. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) |
Theorem | coprmdvdsOLD 15367 | If an integer divides the product of two integers and is coprime to one of them, then it divides the other. (Contributed by Paul Chapman, 22-Jun-2011.) Obsolete version of coprmdvds 15366 as of 10-Jul-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
Theorem | coprmdvds2 15368 | If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.) |
Theorem | mulgcddvds 15369 | One half of rpmulgcd2 15370, which does not need the coprimality assumption. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | rpmulgcd2 15370 | If is relatively prime to , then the GCD of with is the product of the GCDs with and respectively. (Contributed by Mario Carneiro, 2-Jul-2015.) |
Theorem | qredeq 15371 | Two equal reduced fractions have the same numerator and denominator. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | qredeu 15372* | Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.) |
Theorem | rpmul 15373 | If is relatively prime to and to , it is also relatively prime to their product. (Contributed by Mario Carneiro, 24-Feb-2014.) (Proof shortened by Mario Carneiro, 2-Jul-2015.) |
Theorem | rpdvds 15374 | If is relatively prime to then it is also relatively prime to any divisor of . (Contributed by Mario Carneiro, 19-Jun-2015.) |
Theorem | coprmprod 15375* | The product of the elements of a sequence of pairwise coprime positive integers is coprime to a positive integer which is coprime to all integers of the sequence. (Contributed by AV, 18-Aug-2020.) |
Theorem | coprmproddvdslem 15376* | Lemma for coprmproddvds 15377: Induction step. (Contributed by AV, 19-Aug-2020.) |
Theorem | coprmproddvds 15377* | If a positive integer is divisible by each element of a set of pairwise coprime positive integers, then it is divisible by their product. (Contributed by AV, 19-Aug-2020.) |
Theorem | congr 15378* | Definition of congruence by integer multiple (see ProofWiki "Congruence (Number Theory)", 11-Jul-2021, https://proofwiki.org/wiki/Definition:Congruence_(Number_Theory)): An integer is congruent to an integer modulo if their difference is a multiple of . See also the definition in [ApostolNT] p. 104: "... is congruent to modulo , and we write (mod ) if divides the difference ", or Wikipedia "Modular arithmetic - Congruence", https://en.wikipedia.org/wiki/Modular_arithmetic#Congruence, 11-Jul-2021,: "Given an integer n > 1, called a modulus, two integers are said to be congruent modulo n, if n is a divisor of their difference (i.e., if there is an integer k such that a-b = kn)". (Contributed by AV, 11-Jul-2021.) |
Theorem | divgcdcoprm0 15379 | Integers divided by gcd are coprime. (Contributed by AV, 12-Jul-2021.) |
Theorem | divgcdcoprmex 15380* | Integers divided by gcd are coprime (see ProofWiki "Integers Divided by GCD are Coprime", 11-Jul-2021, https://proofwiki.org/wiki/Integers_Divided_by_GCD_are_Coprime): Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their gcd. (Contributed by AV, 12-Jul-2021.) |
Theorem | cncongr1 15381 | One direction of the bicondition in cncongr 15383. Theorem 5.4 in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongr2 15382 | The other direction of the bicondition in cncongr 15383. (Contributed by AV, 11-Jul-2021.) |
Theorem | cncongr 15383 | Cancellability of Congruences (see ProofWiki "Cancellability of Congruences, https://proofwiki.org/wiki/Cancellability_of_Congruences, 10-Jul-2021): Two products with a common factor are congruent modulo a positive integer iff the other factors are congruent modulo the integer divided by the greates common divisor of the integer and the common factor. See also Theorem 5.4 "Cancellation law" in [ApostolNT] p. 109. (Contributed by AV, 13-Jul-2021.) |
Theorem | cncongrcoprm 15384 | Corollary 1 of Cancellability of Congruences: Two products with a common factor are congruent modulo an integer being coprime to the common factor iff the other factors are congruent modulo the integer. (Contributed by AV, 13-Jul-2021.) |
Remark: to represent odd prime numbers, i.e., all prime numbers except , the idiom is used. It is a little bit shorter than . Both representations can be converted into each other by eldifsn 4317. | ||
Syntax | cprime 15385 | Extend the definition of a class to include the set of prime numbers. |
Definition | df-prm 15386* | Define the set of prime numbers. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm 15387* | The predicate "is a prime number". A prime number is a positive integer with exactly two positive divisors. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | prmnn 15388 | A prime number is a positive integer. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | prmz 15389 | A prime number is an integer. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Jonathan Yan, 16-Jul-2017.) |
Theorem | prmssnn 15390 | The prime numbers are a subset of the positive integers. (Contributed by AV, 22-Jul-2020.) |
Theorem | prmex 15391 | The set of prime numbers exists. (Contributed by AV, 22-Jul-2020.) |
Theorem | 1nprm 15392 | 1 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
Theorem | 1idssfct 15393* | The positive divisors of a positive integer include 1 and itself. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2lem 15394* | Lemma for isprm2 15395. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | isprm2 15395* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only positive divisors are 1 and itself. Definition in [ApostolNT] p. 16. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | isprm3 15396* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 with no divisors strictly between 1 and itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | isprm4 15397* | The predicate "is a prime number". A prime number is an integer greater than or equal to 2 whose only divisor greater than or equal to 2 is itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
Theorem | prmind2 15398* | A variation on prmind 15399 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | prmind 15399* | Perform induction over the multiplicative structure of . If a property holds for the primes and and is preserved under multiplication, then it holds for every positive integer. (Contributed by Mario Carneiro, 20-Jun-2015.) |
Theorem | dvdsprime 15400 | If divides a prime, then is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.) |
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