Number of integral values of M

Question

If $\log_3M=a_1+b_1$ and $\log_5 M=a_2+b_2$, where $a_1,a_2$ are natural numbers and $b_1,b_2 \in [0,1)$. If $a_1a_2=6$, then find the number of integral values of M. What so I do in the problem. I don't think anything can be computed. It looks like an estimation problem. How to approach? Thanks.

Answer 1

Hint:

Since $a_1a_2=6$ and $a_2<a_1$, either $a_1=3$ and $a_2=2$, or $a_1=6$ and $a_2=1$.

1) If $a_1=3$ and $a_2=2$, we have $27\le M<81$ and $25\le M<125$.

2) If $a_1=6$ and $a_2=1$, then $3^6\le M<3^7$ and $5\le M<25$.