Euclid’s lemma

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.


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