If a natural number to the third power has exactly 100 divisors, then the natural number itself has at least 100 divisors______ A divisor

If a natural number to the third power has exactly 100 divisors, then the natural number itself has at least 100 divisors______ A divisor

Let this natural number be a, then the prime factor of a decomposition is a = A1B1 × a2b2 × a3b3 × Then A3 = a13b1 × a23b2 × a33b3 × The number of divisors of A3 is 100. According to the theorem of sum of divisors, we can get: (3b 1 + 1) × (3b 2 + 1) + (3b 3 + 1) × X (3bn + 1) = 100, and 100 = 2 × 2 × 5 × 5, because B1, B2, B3 When B1 = 3, B2 = 3, n = 2, the number of divisors of a is: (3 + 1) × (3 + 1) = 16, (2) when B1 = 33, n = 1, the number of divisors of a is: 33 + 1 = 34. A: to sum up, the natural number itself has at least 16 divisors. So the answer is: 16