A divisor of a power of ten

A divisor is chosen uniformly at random from all positive divisors of 109910^{99}. What is the probability it is divisible by 108010^{80}?

Show hints (2)+
  1. Divisors are 2a5b2^a5^b, 0a,b990\le a,b\le99: 1002100^2 of them.
  2. Need a,b80a,b\ge80 (20 values each): 202/100220^2/100^2.

Answer

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0.04

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Asked at: Jane Street, Citadel

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