Euclidean division is based on the following result, which is sometimes called Euclid's division lemma.
Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r

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Although "Euclidean division" is named after Euclid, it seems that he did not know the existence and uniqueness theorem, and that the only

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Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over.
This can be confirmed

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• If a = 7 and b = 3, then q = 2 and r = 1, since 7 = 3 × 2 + 1.
• If a = 7 and b = −3, then q = −2 and r = 1, since 7 = −3 × (−2) + 1.
• If a = −7 and b = 3, then q = −3 and r = 2, since −7 = 3 × (−3) + 2.

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In general, an existence proof does not provide an algorithm for computing the existing quotient and remainder, but the above proof does immediately provide an algorithm (see

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The Euclidean division admits a number of variants, some of which are listed below.
Other intervals for the remainder
In Euclidean division with d as divisor, the remainder is supposed to

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The following proof of the division theorem relies on the fact that a decreasing sequence of non-negative integers stops eventually. It is separated into two parts: one for existence and another for uniqueness of and . Other proofs use the

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