What is the GCD of a and b?

1. The only common prime factor of a and b is even.
2. b/p (where p is a prime number and b is divisible by p) and a don’t have a common prime factor.

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Hi mba878

From the first statement, we learn that the common prime factor is 2. But we don't know whether 2 or 2^2 or 2^3.... is common between a and b.
The second statement says b/p and a don't have a common prime factor, so p (=2^1) must've been the common factor of a and b.

Essentially when we combine the two statements, we learn that 2 is the only common prime factor and only the first power of 2 is a factor of b.
If that weren't the case, b/2 would still contain at least one power of 2 and there would be a common factor between b/2 and a, which goes against statement 2.

As an example, take a = 6 and b = 10.
Their only common prime factor is 2, which is even.
And 6 and 10/2 (=5) don't have a common prime factor.
So we can easily say that 2 is the GCD here.

Hi Shree9975

Let's say b/p = k, where k is an integer since b is divisible by p as per the question.

Statement 2 means k and a don't have any prime factor in common, e.g. k can be 6 and a can be 35.
Here, the prime factors of 6 are 2 and 3 whereas those of 35 are 5 and 7.



From the first statement, we learn that the common prime factor of a and b is 2.

From the second, we know that a and b/p don't have a common prime factor. So either a and b don't have a common factor or they can have p as a common factor. So, either the GCD is 1 or p.

When we combine these two statements, we know that a and b have only one common prime factor 2, so p = 2.

Let us know if that answers your query.