This seems to be a simple problem, but I stuck in the middle. I started with the formula $(a,b)[a,b] = ab$, and got $(a, 60)1260 = 60a$, which is $12(a,60)=a$. Then what should I do?
The least common multiple (LCM) of two positive integers is 1260. If one of the numbers is 60, what is the smallest possible value of the other number?
$$1260=2^2\cdot 3^2 \cdot 5 \cdot 7$$
$$60=2^2\cdot 3 \cdot 5$$ Therefore we need the smallest number divisible by seven and nine, which is $7\cdot9=63$.