What is the smallest number that has 100 different positive integral factors including one and itself?

The smallest number that has exactly 100 different positive integral factors is 2^6 × 3^4. To find the number of factors of a number based on its prime factorization, you can use the formula (e1 + 1)(e2 + 1)...(en + 1), where e1, e2, ..., en are the exponents in the prime factorization. For 2^6 × 3^4, the number of factors is (6 + 1)(4 + 1) = 7 × 5 = 35, which is less than 100, so a different combination of prime factorization would be needed to achieve exactly 100 factors. The correct combination resulting in 100 factors is 2^6 × 3^2 × 5^1, yielding the smallest number, which is 2^6 × 3^2 × 5 = 8640.