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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | gcdadd 10901 | The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.) |
Theorem | gcdid 10902 | The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | gcd1 10903 | The gcd of a number with 1 is 1. Theorem 1.4(d)1 in [ApostolNT] p. 16. (Contributed by Mario Carneiro, 19-Feb-2014.) |
Theorem | gcdabs 10904 | The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | gcdabs1 10905 | of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcdabs2 10906 | of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | modgcd 10907 | The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | 1gcd 10908 | The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | 6gcd4e2 10909 | The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: and . (Contributed by AV, 27-Aug-2020.) |
Theorem | bezoutlemnewy 10910* | Lemma for Bézout's identity. The is-bezout predicate holds for . (Contributed by Jim Kingdon, 6-Jan-2022.) |
Theorem | bezoutlemstep 10911* | Lemma for Bézout's identity. This is the induction step for the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.) |
Theorem | bezoutlemmain 10912* | Lemma for Bézout's identity. This is the main result which we prove by induction and which represents the application of the Extended Euclidean algorithm. (Contributed by Jim Kingdon, 30-Dec-2021.) |
Theorem | bezoutlema 10913* | Lemma for Bézout's identity. The is-bezout condition is satisfied by . (Contributed by Jim Kingdon, 30-Dec-2021.) |
Theorem | bezoutlemb 10914* | Lemma for Bézout's identity. The is-bezout condition is satisfied by . (Contributed by Jim Kingdon, 30-Dec-2021.) |
Theorem | bezoutlemex 10915* | Lemma for Bézout's identity. Existence of a number which we will later show to be the greater common divisor and its decomposition into cofactors. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jan-2022.) |
Theorem | bezoutlemzz 10916* | Lemma for Bézout's identity. Like bezoutlemex 10915 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
Theorem | bezoutlemaz 10917* | Lemma for Bézout's identity. Like bezoutlemzz 10916 but where ' A ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
Theorem | bezoutlembz 10918* | Lemma for Bézout's identity. Like bezoutlemaz 10917 but where ' B ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
Theorem | bezoutlembi 10919* | Lemma for Bézout's identity. Like bezoutlembz 10918 but the greatest common divisor condition is a biconditional, not just an implication. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
Theorem | bezoutlemmo 10920* | Lemma for Bézout's identity. There is at most one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) |
Theorem | bezoutlemeu 10921* | Lemma for Bézout's identity. There is exactly one nonnegative integer meeting the greatest common divisor condition. (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) |
Theorem | bezoutlemle 10922* | Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the largest number which divides both and . (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) |
Theorem | bezoutlemsup 10923* | Lemma for Bézout's identity. The number satisfying the greatest common divisor condition is the supremum of divisors of both and . (Contributed by Mario Carneiro and Jim Kingdon, 9-Jan-2022.) |
Theorem | dfgcd3 10924* | Alternate definition of the operator. (Contributed by Jim Kingdon, 31-Dec-2021.) |
Theorem | bezout 10925* |
Bézout's identity: For any integers and , there are
integers such that
. This
is Metamath 100 proof #60.
The proof is constructive, in the sense that it applies the Extended Euclidian Algorithm to constuct a number which can be shown to be and which satisfies the rest of the theorem. In the presence of excluded middle, it is common to prove Bézout's identity by taking the smallest number which satisfies the Bézout condition, and showing it is the greatest common divisor. But we do not have the ability to show that number exists other than by providing a way to determine it. (Contributed by Mario Carneiro, 22-Feb-2014.) |
Theorem | dvdsgcd 10926 | An integer which divides each of two others also divides their gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 30-May-2014.) |
Theorem | dvdsgcdb 10927 | Biconditional form of dvdsgcd 10926. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | dfgcd2 10928* | Alternate definition of the operator, see definition in [ApostolNT] p. 15. (Contributed by AV, 8-Aug-2021.) |
Theorem | gcdass 10929 | Associative law for operator. Theorem 1.4(b) in [ApostolNT] p. 16. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | mulgcd 10930 | Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.) |
Theorem | absmulgcd 10931 | Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | mulgcdr 10932 | Reverse distribution law for the operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcddiv 10933 | Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcdmultiple 10934 | The GCD of a multiple of a number is the number itself. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcdmultiplez 10935 | Extend gcdmultiple 10934 so can be an integer. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | gcdzeq 10936 | A positive integer is equal to its gcd with an integer if and only if divides . Generalization of gcdeq 10937. (Contributed by AV, 1-Jul-2020.) |
Theorem | gcdeq 10937 | is equal to its gcd with if and only if divides . (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by AV, 8-Aug-2021.) |
Theorem | dvdssqim 10938 | Unidirectional form of dvdssq 10945. (Contributed by Scott Fenton, 19-Apr-2014.) |
Theorem | dvdsmulgcd 10939 | Relationship between the order of an element and that of a multiple. (a divisibility equivalent). (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Theorem | rpmulgcd 10940 | If and are relatively prime, then the GCD of and is the GCD of and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | rplpwr 10941 | If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | rppwr 10942 | If and are relatively prime, then so are and . (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | sqgcd 10943 | Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | dvdssqlem 10944 | Lemma for dvdssq 10945. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | dvdssq 10945 | Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Theorem | bezoutr 10946 | Partial converse to bezout 10925. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | bezoutr1 10947 | Converse of bezout 10925 for when the greater common divisor is one (sufficient condition for relative primality). (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Theorem | nn0seqcvgd 10948* | A strictly-decreasing nonnegative integer sequence with initial term reaches zero by the th term. Deduction version. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | ialgrlem1st 10949 | Lemma for ialgr0 10951. Expressing algrflemg 5954 in a form suitable for theorems such as iseq1 9805 or iseqfcl 9808. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | ialgrlemconst 10950 | Lemma for ialgr0 10951. Closure of a constant function, in a form suitable for theorems such as iseq1 9805 or iseqfcl 9808. (Contributed by Jim Kingdon, 22-Jul-2021.) |
Theorem | ialgr0 10951 | The value of the algorithm iterator at is the initial state . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | ialgrf 10952 |
An algorithm is a step function on a state space .
An algorithm acts on an initial state by
iteratively applying
to give , , and so
on. An algorithm is said to halt if a fixed point of is reached
after a finite number of iterations.
The algorithm iterator "runs" the algorithm so that is the state after iterations of on the initial state . Domain and codomain of the algorithm iterator . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | ialgrp1 10953 | The value of the algorithm iterator at . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Theorem | ialginv 10954* | If is an invariant of , its value is unchanged after any number of iterations of . (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | ialgcvg 10955* |
One way to prove that an algorithm halts is to construct a countdown
function whose value is guaranteed to decrease for
each iteration of until it reaches . That is, if
is not a fixed point of , then
.
If is a countdown function for algorithm , the sequence reaches after at most steps, where is the value of for the initial state . (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | algcvgblem 10956 | Lemma for algcvgb 10957. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | algcvgb 10957 | Two ways of expressing that is a countdown function for algorithm . The first is used in these theorems. The second states the condition more intuitively as a conjunction: if the countdown function's value is currently nonzero, it must decrease at the next step; if it has reached zero, it must remain zero at the next step. (Contributed by Paul Chapman, 31-Mar-2011.) |
Theorem | ialgcvga 10958* | The countdown function remains after steps. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | ialgfx 10959* | If reaches a fixed point when the countdown function reaches , remains fixed after steps. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | eucalgval2 10960* | The value of the step function for Euclid's Algorithm on an ordered pair. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | eucalgval 10961* |
Euclid's Algorithm eucialg 10966 computes the greatest common divisor of two
nonnegative integers by repeatedly replacing the larger of them with its
remainder modulo the smaller until the remainder is 0.
The value of the step function for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | eucalgf 10962* | Domain and codomain of the step function for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Theorem | eucalginv 10963* | The invariant of the step function for Euclid's Algorithm is the operator applied to the state. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.) |
Theorem | eucalglt 10964* | The second member of the state decreases with each iteration of the step function for Euclid's Algorithm. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 29-May-2014.) |
Theorem | eucialgcvga 10965* | Once Euclid's Algorithm halts after steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.) |
Theorem | eucialg 10966* |
Euclid's Algorithm computes the greatest common divisor of two
nonnegative integers by repeatedly replacing the larger of them with
its remainder modulo the smaller until the remainder is 0. Theorem
1.15 in [ApostolNT] p. 20.
Upon halting, the 1st member of the final state is equal to the gcd of the values comprising the input state . This is Metamath 100 proof #69 (greatest common divisor algorithm). (Contributed by Jim Kingdon, 11-Jan-2022.) |
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." In this section, an operation calculating the least common multiple of two integers (df-lcm 10968). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention. | ||
Syntax | clcm 10967 | Extend the definition of a class to include the least common multiple operator. |
lcm | ||
Definition | df-lcm 10968* | Define the lcm operator. For example, lcm . (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmmndc 10969 | Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.) |
DECID | ||
Theorem | lcmval 10970* | 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 10864 and gcdval 10876. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
lcm inf | ||
Theorem | lcmcom 10971 | The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcm0val 10972 | The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 10971 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmn0val 10973* | 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 10974* | Lemma for lcmn0cl 10975 and dvdslcm 10976. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm | ||
Theorem | lcmn0cl 10975 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | dvdslcm 10976 | The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmledvds 10977 | 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 10978 | The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmcl 10979 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | gcddvdslcm 10980 | The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmneg 10981 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | neglcm 10982 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm lcm | ||
Theorem | lcmabs 10983 | 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 10984 | Lemma for lcmgcd 10985 and lcmdvds 10986. Prove them for positive , , and . (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
lcm lcm | ||
Theorem | lcmgcd 10985 |
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 or of Bézout's identity bezout 10925; 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 10972 to show it applies when either or both inputs are zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmdvds 10986 | The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmid 10987 | The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcm1 10988 | The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.) |
lcm | ||
Theorem | lcmgcdnn 10989 | 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 10990 | 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 10991 | Biconditional form of lcmdvds 10986. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
lcm | ||
Theorem | lcmass 10992 | 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 10993 | The least common multiple of three and two is six. This proof does not use the property of 2 and 3 being prime. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 27-Aug-2020.) |
lcm | ||
Theorem | 6lcm4e12 10994 | The least common multiple of six and four is twelve. (Contributed 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 10995. The negation, i.e. two integers are not coprime, can be expressed either by , see ncoprmgcdne1b 10996, or equivalently by , see ncoprmgcdgt1b 10997. A proof of Euclid's lemma based on coprimality is provided in coprmdvds 10999 (as opposed to Euclid's lemma for primes). | ||
Theorem | coprmgcdb 10995* | 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 10996* | 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 10997* | 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 10998 | 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 10999 | 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. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.) |
Theorem | coprmdvds2 11000 | If an integer is divisible by two coprime integers, then it is divisible by their product. (Contributed by Mario Carneiro, 24-Feb-2014.) |
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