AMC 12 Mock Exam 5 Problem 10

The beacons flash together exactly at the common multiples of $a$ and $b$, so the condition is $\operatorname{lcm}(a,b)=72$. Write $72=2^3\cdot3^2$; both $a$ and $b$ divide $72$. For the prime $2$, the exponents $(i,j)$ of $2$ in $a$ and $b$ satisfy $0\le i,j\le3$ and $\max(i,j)=3$: there are $4^2-3^2=7$ such pairs. For the prime $3$, similarly $3^2-2^2=5$ pairs. The choices for different primes are independent, so the answer is $7\cdot5=\boxed{35}$.