Find the smallest positive integer such that it has exactly 100 different positive integer divisors, including 1 and the number itself.

The number of divisors of $$N=p_1^{e_1}p_2^{e_2}\cdots$$ is $$(e_1+1)(e_2+1)\cdots$$, and 100 factors as $$100 = 5\times 5\times 2\times 2$$, corresponding to exponents 4, 4, 1, 1. To minimize N, the largest exponents should go on the smallest primes, giving $$N = 2^4 \times 3^4 \times 5 \times 7$$. Computing this product gives $$16 \times 81 \times 5 \times 7 = 45360$$.