Find the total number of divisors of 1728 (including 1 and 1728).

Concept: For a number N with prime factorization N = pa × qb × rc × … (p, q, r distinct primes), the total number of positive divisors of N — including 1 and N itself — equals (a + 1)(b + 1)(c + 1) …, one factor for every distinct prime in the factorization.

Application:

1. Find the prime factorization of 1728 by dividing out the smallest prime repeatedly: dividing by 2 six times in a row (1728 → 864 → 432 → 216 → 108 → 54 → 27) leaves 27, which further divides evenly by 3 three times (27 → 9 → 3 → 1).

2. So 1728 = 26 × 33, with exponents 6 (for the prime 2) and 3 (for the prime 3).

3. Apply the divisor-count rule above: total divisors = (6 + 1)(3 + 1) = 7 × 4 = 28.

Cross-check: Multiplying the factorization back confirms it: 26 = 64 and 33 = 27, and 64 × 27 = 1728, so the factorization used is correct.

Result: 1728 has 28 total divisors (including 1 and 1728), matching this option.