If 2352 = 2x × 3y × 7z, then the value of x + y + z is

Concept: The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has a UNIQUE prime factorization — a product of primes raised to whole-number exponents. Once a number is written in this form, its exponents can be read off directly.

1. Divide 2352 by 2 repeatedly until the quotient turns odd: 2352 → 1176 → 588 → 294 → 147 — four divisions, so 24 is a factor and 147 is odd.

2. Factor the odd part 147: 147 ÷ 3 = 49 (49 is not divisible by 3 again), so 31 is a factor.

3. Factor 49: 49 = 7 × 7 = 72.

4. Combine: 2352 = 24 × 31 × 72. Matching against 2x × 3y × 7z gives x = 4, y = 1, z = 2.

5. Sum the exponents: x + y + z = 4 + 1 + 2 = 7.

Cross-check: multiply back — 24 × 3 × 72 = 16 × 3 × 49 = 16 × 147 = 2352, confirming the factorization (and hence the exponent sum) is correct.