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Type | Label | Description |
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Statement | ||
Theorem | gcdeq0 11701 | The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | gcdn0gt0 11702 | The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | gcd0id 11703 | The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | gcdid0 11704 | The gcd of an integer and 0 is the integer's absolute value. Theorem 1.4(d)2 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | nn0gcdid0 11705 | The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | gcdneg 11706 |
Negating one operand of the ![]() |
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Theorem | neggcd 11707 |
Negating one operand of the ![]() |
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Theorem | gcdaddm 11708 |
Adding a multiple of one operand of the ![]() |
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Theorem | gcdadd 11709 | The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.) |
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Theorem | gcdid 11710 | The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | gcd1 11711 | 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.) |
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Theorem | gcdabs 11712 | The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | gcdabs1 11713 |
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Theorem | gcdabs2 11714 |
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Theorem | modgcd 11715 | The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | 1gcd 11716 | The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | gcdmultipled 11717 |
The greatest common divisor of a nonnegative integer ![]() ![]() |
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Theorem | dvdsgcdidd 11718 | The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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Theorem | 6gcd4e2 11719 |
The greatest common divisor of six and four is two. To calculate this
gcd, a simple form of Euclid's algorithm is used:
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Theorem | bezoutlemnewy 11720* |
Lemma for Bézout's identity. The is-bezout predicate holds for
![]() ![]() ![]() ![]() ![]() |
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Theorem | bezoutlemstep 11721* | Lemma for Bézout's identity. This is the induction step for the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.) |
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Theorem | bezoutlemmain 11722* | 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.) |
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Theorem | bezoutlema 11723* |
Lemma for Bézout's identity. The is-bezout condition is
satisfied by ![]() |
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Theorem | bezoutlemb 11724* |
Lemma for Bézout's identity. The is-bezout condition is
satisfied by ![]() |
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Theorem | bezoutlemex 11725* | 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.) |
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Theorem | bezoutlemzz 11726* | Lemma for Bézout's identity. Like bezoutlemex 11725 but where ' z ' is any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
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Theorem | bezoutlemaz 11727* | Lemma for Bézout's identity. Like bezoutlemzz 11726 but where ' A ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
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Theorem | bezoutlembz 11728* | Lemma for Bézout's identity. Like bezoutlemaz 11727 but where ' B ' can be any integer, not just a nonnegative one. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
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Theorem | bezoutlembi 11729* | Lemma for Bézout's identity. Like bezoutlembz 11728 but the greatest common divisor condition is a biconditional, not just an implication. (Contributed by Mario Carneiro and Jim Kingdon, 8-Jan-2022.) |
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Theorem | bezoutlemmo 11730* | 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.) |
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Theorem | bezoutlemeu 11731* | 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.) |
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Theorem | bezoutlemle 11732* |
Lemma for Bézout's identity. The number satisfying the
greatest common divisor condition is the largest number which
divides both ![]() ![]() |
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Theorem | bezoutlemsup 11733* |
Lemma for Bézout's identity. The number satisfying the
greatest common divisor condition is the supremum of divisors of
both ![]() ![]() |
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Theorem | dfgcd3 11734* |
Alternate definition of the ![]() |
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Theorem | bezout 11735* |
Bézout's identity: For any integers ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
The proof is constructive, in the sense that it applies the Extended
Euclidian Algorithm to constuct a number which can be shown to be
|
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Theorem | dvdsgcd 11736 | 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.) |
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Theorem | dvdsgcdb 11737 | Biconditional form of dvdsgcd 11736. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | dfgcd2 11738* |
Alternate definition of the ![]() |
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Theorem | gcdass 11739 |
Associative law for ![]() |
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Theorem | mulgcd 11740 | Distribute multiplication by a nonnegative integer over gcd. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 30-May-2014.) |
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Theorem | absmulgcd 11741 | Distribute absolute value of multiplication over gcd. Theorem 1.4(c) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) |
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Theorem | mulgcdr 11742 |
Reverse distribution law for the ![]() |
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Theorem | gcddiv 11743 | Division law for GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | gcdmultiple 11744 | 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.) |
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Theorem | gcdmultiplez 11745 |
Extend gcdmultiple 11744 so ![]() |
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Theorem | gcdzeq 11746 |
A positive integer ![]() ![]() ![]() ![]() |
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Theorem | gcdeq 11747 |
![]() ![]() ![]() ![]() |
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Theorem | dvdssqim 11748 | Unidirectional form of dvdssq 11755. (Contributed by Scott Fenton, 19-Apr-2014.) |
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Theorem | dvdsmulgcd 11749 | Relationship between the order of an element and that of a multiple. (a divisibility equivalent). (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | rpmulgcd 11750 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rplpwr 11751 |
If ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | rppwr 11752 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | sqgcd 11753 | Square distributes over GCD. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | dvdssqlem 11754 | Lemma for dvdssq 11755. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | dvdssq 11755 | Two numbers are divisible iff their squares are. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | bezoutr 11756 | Partial converse to bezout 11735. Existence of a linear combination does not set the GCD, but it does upper bound it. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
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Theorem | bezoutr1 11757 | Converse of bezout 11735 for when the greater common divisor is one (sufficient condition for relative primality). (Contributed by Stefan O'Rear, 23-Sep-2014.) |
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Theorem | nn0seqcvgd 11758* |
A strictly-decreasing nonnegative integer sequence with initial term
![]() ![]() |
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Theorem | ialgrlem1st 11759 | Lemma for ialgr0 11761. Expressing algrflemg 6135 in a form suitable for theorems such as seq3-1 10264 or seqf 10265. (Contributed by Jim Kingdon, 22-Jul-2021.) |
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Theorem | ialgrlemconst 11760 | Lemma for ialgr0 11761. Closure of a constant function, in a form suitable for theorems such as seq3-1 10264 or seqf 10265. (Contributed by Jim Kingdon, 22-Jul-2021.) |
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Theorem | ialgr0 11761 |
The value of the algorithm iterator ![]() ![]() ![]() |
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Theorem | algrf 11762 |
An algorithm is a step function ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
The algorithm iterator
Domain and codomain of the algorithm iterator |
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Theorem | algrp1 11763 |
The value of the algorithm iterator ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | alginv 11764* |
If ![]() ![]() ![]() |
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Theorem | algcvg 11765* |
One way to prove that an algorithm halts is to construct a countdown
function ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
If |
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Theorem | algcvgblem 11766 | Lemma for algcvgb 11767. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | algcvgb 11767 |
Two ways of expressing that ![]() ![]() |
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Theorem | algcvga 11768* |
The countdown function ![]() ![]() ![]() |
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Theorem | algfx 11769* |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | eucalgval2 11770* |
The value of the step function ![]() |
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Theorem | eucalgval 11771* |
Euclid's Algorithm eucalg 11776 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 |
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Theorem | eucalgf 11772* |
Domain and codomain of the step function ![]() |
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Theorem | eucalginv 11773* |
The invariant of the step function ![]() ![]() |
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Theorem | eucalglt 11774* |
The second member of the state decreases with each iteration of the step
function ![]() |
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Theorem | eucalgcvga 11775* |
Once Euclid's Algorithm halts after ![]() |
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Theorem | eucalg 11776* |
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 |
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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 11778). The definition is valid for all integers, including negative integers and 0, obeying the above mentioned convention. | ||
Syntax | clcm 11777 | Extend the definition of a class to include the least common multiple operator. |
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Definition | df-lcm 11778* |
Define the lcm operator. For example, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | lcmmndc 11779 | Decidablity lemma used in various proofs related to lcm. (Contributed by Jim Kingdon, 21-Jan-2022.) |
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Theorem | lcmval 11780* |
Value of the lcm operator. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | lcmcom 11781 | The lcm operator is commutative. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
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Theorem | lcm0val 11782 | The value, by convention, of the lcm operator when either operand is 0. (Use lcmcom 11781 for a left-hand 0.) (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
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Theorem | lcmn0val 11783* | The value of the lcm operator when both operands are nonzero. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.) |
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Theorem | lcmcllem 11784* | Lemma for lcmn0cl 11785 and dvdslcm 11786. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.) |
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Theorem | lcmn0cl 11785 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | dvdslcm 11786 | The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcmledvds 11787 | 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.) |
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Theorem | lcmeq0 11788 | The lcm of two integers is zero iff either is zero. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcmcl 11789 | Closure of the lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | gcddvdslcm 11790 | The greatest common divisor of two numbers divides their least common multiple. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcmneg 11791 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | neglcm 11792 | Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcmabs 11793 | The lcm of two integers is the same as that of their absolute values. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcmgcdlem 11794 |
Lemma for lcmgcd 11795 and lcmdvds 11796. Prove them for positive ![]() ![]() ![]() |
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Theorem | lcmgcd 11795 |
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 11735; see e.g.
https://proofwiki.org/wiki/Product_of_GCD_and_LCM 11735 and
https://math.stackexchange.com/a/470827 11735. This proof uses the latter to
first confirm it for positive integers |
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Theorem | lcmdvds 11796 | The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcmid 11797 | The lcm of an integer and itself is its absolute value. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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Theorem | lcm1 11798 | The lcm of an integer and 1 is the absolute value of the integer. (Contributed by AV, 23-Aug-2020.) |
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Theorem | lcmgcdnn 11799 | 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.) |
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Theorem | lcmgcdeq 11800 | Two integers' absolute values are equal iff their least common multiple and greatest common divisor are equal. (Contributed by Steve Rodriguez, 20-Jan-2020.) |
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