26th Putnam 1965

How many positive integers divide at least one of 10⁴⁰ and 20³⁰?

Solution

Answer: 2301.

Factors of 10⁴⁰ have the form 2^m5ⁿ with 0 <= m, n <= 40. So there are 41² = 1681 such factors. Factors of 20³⁰ = 2⁶⁰5³⁰ not dividing 10⁴⁰ have the form 2^m5ⁿ with 41 ≤ m ≤ 60 and 0 ≤ n ≤ 30, so there are 20.31 = 620 such factors.

4th Putnam 1941

© John Scholes
jscholes@kalva.demon.co.uk
3 Nov 1999