# Euclid’s lemma

• Difficulty Level : Easy
• Last Updated : 19 Nov, 2016

We are given two numbers x and y. We know that a number p divides their product. Can we say for sure that p also divides one of them?

The answer is no. For example, consider x = 15, y = 6 and p = 9. p divides the product 15*6, but doesn’t divide any of them.

What if p is prime?
Euclid’s lemma states that if a prime p divides the product of two numbers (x*y), it must divide at least one of those numbers.

For example x = 15, y = 6 and p = 5. p divides the product 15*6, it also divides 15.

The idea is simple, since p is prime, it cannot be factorized. So it must either be completely present in x or in y.

Generalization of Euclid’s lemma:
If p divides x*y and p is relatively prime to x, then p must divide y. In the above example, 5 is relatively prime to 6, therefore it must divide 15.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

My Personal Notes arrow_drop_up