Question

If `a` and `b` are coprime and `c` is a factor of `a`, then prove that `b` and `c` too are coprime?

Answer 1

Please see below.

Explanation:

Numbers whose GCD is `1` are known as coprime or relatively prime.

If two numbers are prime numbers, they will be coprime. But even if there are two composite numbers, let us say `25` and `36`, as there is no common factor between them, they are coprime and their GCD is `1`.

Now as GCD of `a` and `b` is `1`, there is no common factor between them.

We have `c`, which is a factor of `a`, but as there is no common factor between `a` and `b`,

there will not be a common factor between `b` and `c`, as otherwise this common factor would have been GCD of `a` and `b`, (as it divides both).

Hence, GCD of `b` and `c` too is `1`.

As an example, GCD of `36` and `25` is `1`, but though `12` is a factor of `36`, GCD of `12` and `25` too is `1`