Answer: 634

Small Hint:

Write every divisor as $2^a5^b$

Big Hint:

Count exponent pairs with $a\geq88$ and $b\geq88$ and reduce the resulting probability

Solution:

Every divisor is $2^a5^b$ with $0\leq a\leq99$ and $0\leq b\leq99,$ so there are $100^2=10000$ divisors. A multiple of $10^{88}$ requires $88\leq a\leq99$ and $88\leq b\leq99,$ giving $12^2=144$ divisors. The probability is $\frac{144}{10000}=\frac{9}{625},$ and $m+n=9+625=634.$