How to find out common factors from factorization of 2 different numbers?

Hey,

After factorization of 2 different numbers(i.e getting the factors of number Y and X), look out for the common prime factors, if there exists common prime factor, then look out for the exponent of the prime factor

I think we should first ask ourselves what a factor is. Consider the number 24. In some intuitive way, we understand that 8\cdot 3=24, so 8 would be a factor of 24 as well as 3, because we can multiply both integers, by some other integer and get 24.

Now, consider 6 and 30. The first thing we always need to do is prime factorization. So 6=2\cdot 3 and 30=2\cdot 3 \cdot 5 What primes do you see in common? 2 and 3, so 2 is a common factor as well as 3. Is that all the common factors there are? No. Notice we only listed prime factors. But what about 3\cdot 2=6? That is a common factor as well.

Now, consider two prime numbers, say 3 and 11. Do they share any common factors? Hint: Recall the definition of a prime number.

They have only 1 common as a factor