Problem:
How many positive factors does 23,232 have?
Answer Choices:
A. 9
B. 12
C. 28
D. 36
E. 42
Solution:
Begin by finding the prime factorization of 23232. To do this, we repeatedly factor out the smallest prime factor from the number, a process that terminates when the number is a prime number. This process is outlined below:
2323211616580829041452726363121=2×11616=2×5808=2×2904=2×1452=2×726=2×363=3×121=11×11
Thus, the prime factorization of 23232 is:
23232=26×3×112
An arbitrary factor of 23232 can be created by taking the product of any number of prime factors. More explicitly, 23232 can be represented as:
23232=p1e1p2e2⋯pmem
where p1,p2,…,pm are prime numbers, and each factor has (e1+1)(e2+1)⋯(em+1) options of prime factorizations to choose from. Thus, the total number of factors is:
(6+1)(1+1)(2+1)=7×2×3=42.
Thus, E is the correct answer.
Answer: E.
The problems on this page are the property of the MAA's American Mathematics Competitions