Problem 18 · 2018 AMC 8
Medium
Number Theory
factorizationprime-test
How many positive factors does 23,232 have?
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Answer: E — 42 factors.
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Hint 1 of 2
Listing all the divisors by hand would be miserable. The escape: every factor is built by choosing how many of each prime to include — so the real question is about the exponents in the prime factorization, not the divisors themselves.
Still stuck? Show hint 2 →
Hint 2 of 2
The technique — the divisor-counting formula: factor the number as primes to powers, then multiply (each exponent + 1). The "+1" is the option of using that prime zero times.
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Approach: prime factorize, multiply (exponent + 1) per prime
- Strip out 2's: 23,232 halves six times down to 363, so 23,232 = 26 · 363. Then 363 = 3 · 121 = 3 · 112. So 23,232 = 26 · 3 · 112.
- Each divisor picks 0–6 copies of 2 (7 choices), 0–1 copies of 3 (2 choices), and 0–2 copies of 11 (3 choices). Multiply the independent choices: 7 · 2 · 3 = 42.
- You'll see it again: "how many factors" is always (exponent+1) multiplied across the primes — and the same choice-counting idea gives the sum of factors too.
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