Euclid division lemma biography of mahatma

A prime divisor of a product divides one of the factors

Not to be confused with Euclid's division lemma, Euclid's theorem, or Euclidean algorithm.

In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:^{[note 1]}

For example, if p = 19, a = , b = , then ab = &#; = , and since this is divisible by 19, the lemma implies that one or both of or must be as well. In fact, = 19 × 7.

The lemma first appeared in Euclid's Elements, and is a fundamental result in elementary number theory.

If the premise of the lemma does not hold, that is, if p is a composite number, its consequent may be either true or false. For example, in the case of p = 10, a = 4, b = 15, composite number 10 divides ab = 4 &#; 15 = 60, but 10 divides neit