Maths Olympiad Prep

Library / /145 of 860

Number theory Difficulty 4.9 AIME Find the answer

Compute the positive integer less than 1000 which has exactly 29 positive proper divisors.

A number or a short expression. Spacing and $ signs are ignored.

Solution

Recall that the number N=p1e1p2e2pkekN=p_{1}^{e_{1}} p_{2}^{e_{2}} \cdots p_{k}^{e_{k}} (where the pip_{i} are distinct primes) has exactly (e1+1)(e2+1)(ek+1)(e_{1}+1)(e_{2}+1) \cdots(e_{k}+1) positive integer divisors including itself. We seek N<1000N<1000 such that this expression is 30. Since 30=23530=2 \cdot 3 \cdot 5, we take e1=1,e2=2,e3=4e_{1}=1, e_{2}=2, e_{3}=4. Then we see that N=513242=720N=5^{1} 3^{2} 4^{2}=720 is satisfactory.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.