unknown at Euclid's time. Bézout's identity states that if and are coprime integers (i.e. they share no common divisors other than 1 and −1) there exist integers and such that Let and be coprime, and assume that . By Bézout's identity, there are and such that Multiply both sides by : The first term on the left is divisible by , and the second term is divisible by , which by hypothesis is divisible by . Therefore their sum, , is also divisible by . By induction The following proof is inspired by Euclid's version of Euclidean algorithm, which proceeds by using only subtractions. Suppose that and that and are coprime (that is, their greatest common divisor is ). One has to prove that divides . Since there is a integer such that Without loss of generality, one can suppose that , , , and are positive, since the divisibility relation is independent from the signs of the involved integers. For proving this by strong induction, we suppose that the result has been proved for all positive lower values of . There are three cases: If , coprimality implies , and divides trivially. If , one has The positive integers and are coprime: their greatest common divisor must divide their sum, and thus divides both and . It results that , by the coprimality hypothesis. So, the conclusion follows from the induction hypothesis, since . Similarly, if one has and the same argument shows that and are coprime. Therefore, one has , and the induction hypothesis implies that divides ; that is, for some integer. So, and, by dividing by , one has Therefore, and by dividing by , one gets the desired result. Proof of Elements Euclid's lemma is proved at the Proposition 30 in Book VII of Euclid's Elements. The original proof is difficult to understand as is, so we quote the commentary from . Proposition 19 If four numbers be proportional, the number produced from the first and fourth is equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers are proportional. Proposition 20 The least numbers of those that have the same ratio with them measures those that have the same ratio the same number of times—the greater the greater and the less