IMO 1986 SL 6

Origin: NET

Problem

Find four positive integers each not exceeding 70000 and each having more than 100 divisors.

Solution

There are five such numbers: 69300 = 22 \cdot 32 \cdot 52 \cdot 7 \cdot 11 : 3 \cdot 3 \cdot 3 \cdot 2 \cdot 2 = 108 divisors; 50400 = 25 \cdot 32 \cdot 52 \cdot 7 : 6 \cdot 3 \cdot 3 \cdot 2 = 108 divisors; 60480 = 26 \cdot 33 \cdot 5 \cdot 7 : 7 \cdot 4 \cdot 2 \cdot 2 = 112 divisors; 55440 = 24 \cdot 32 \cdot 5 \cdot 7 \cdot 11 : 5 \cdot 3 \cdot 2 \cdot 2 \cdot 2 = 120 divisors; 65520 = 24 \cdot 32 \cdot 5 \cdot 7 \cdot 13 : 5 \cdot 3 \cdot 2 \cdot 2 \cdot 2 = 120 divisors.