Problem 12 · 2020 AMC 8
Easy
Number Theory
factorization
For a positive integer n, the factorial notation n! represents the product of the integers from n to 1. For example: 6! = 6 × 5 × 4 × 3 × 2 × 1. What value of N satisfies the following equation?
5! × 9! = 12 × N!
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Answer: A — N = 10.
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Hint 1 of 2
Don't expand 9! into a giant number. The right side has a 9! sitting in it — so leave the 9! alone and shape the left side to match it.
Still stuck? Show hint 2 →
Hint 2 of 2
The key identity is 10! = 10 × 9!. So if you can turn 5! into 12 × 10, the whole left side becomes 12 × 10 × 9! = 12 × 10!.
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Approach: reassemble a factorial instead of multiplying it out
- Factorials “telescope”: n! = n × (n−1)!. So a leftover factor times 9! can rebuild into 10!, 11!, … — keep the 9! intact rather than expanding it.
- 5! = 120 = 12 × 10. So 5! × 9! = 12 × (10 × 9!) = 12 × 10!.
- Matching 12 × N! = 12 × 10! gives N = 10.
- You'll see this again as: when factorials appear in an equation, never compute the big ones — peel digits off one factorial and absorb them into another using n! = n(n−1)!. The arithmetic stays tiny.
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