If \(a\) and \(b\) are positive integers such that \(\gcd(a,b)=210\), \(\mathop{\text{lcm}}[a,b]=210^3\), and \(a , how many possible values are there for \(a\)?

a and b are the same as their GCD and LCM. In other words:

GCD(210, 210^3) = 210, and

LCM(210, 210^3) = 210^3. If "a" stands for 210, then "a" can only take one value of 210. If you double it, say to 420, then the new GCD = 420....and so on.