Suppose a certain student’s ID number M satisfies

gcd(M, 2010) > gcd(M, 271) > 1.

Find all possible values for gcd(M, 2010). Be sure to explain your reasoning. [Note: both 271 and 67 are prime.]

I have gotten most of the way through the question....but will wait to see if anyone else has any insights. It is very possible that I am stuck because one of my steps lead me in the wrong direction...