Find the number of divisors of 1728?

Concept: For a positive integer N written in prime-factorized form N = pa × qb × ... (where p, q, ... are distinct primes, each raised to some power), the total number of positive divisors of N equals (a+1)(b+1)... – the product of each exponent plus one – because every divisor is formed by independently picking, for each prime, an exponent between 0 and its maximum power.

1. Factorize 1728 into primes: dividing repeatedly by 2 gives 1728 = 26 × 27, and 27 = 33, so 1728 = 26 × 33.

2. Read off the exponents in this factorization: 6 for the prime 2, and 3 for the prime 3.

3. Apply the divisor-count formula from the concept above: (6+1) × (3+1) = 7 × 4 = 28.

Cross-check: 1728 = 123 = (22 × 3)3 = 26 × 33, the same factorization used above – confirming the exponents.

So 1728 has 28 positive divisors.