What is the probability that divisor of 1099 is a multiple of 1096?

Key idea: Count all divisors of 10^99 and those that are multiples of 10^96.

• Factor 10^99 as 2^99 · 5^99.

• Any divisor has the form 2^a · 5^b with 0 ≤ a,b ≤ 99, so total divisors = 100 × 100 = 10000.

• For a divisor to be a multiple of 10^96 we need a ≥ 96 and b ≥ 96. Each exponent has choices 96,97,98,99, so 4 choices for a and 4 for b, giving 4 × 4 = 16 favorable divisors.

• Therefore probability = favorable/total = 16/10000 = 1/625

Answer: 1/625