Problem 5 · 2017 AMC 8
Easy
Arithmetic & Operations
factorizationarithmetic-series
What is the value of the expression
1 · 2 · 3 · 4 · 5 · 6 · 7 · 81 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ?
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Answer: B — 1120.
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Hint 1 of 2
Don't multiply the giant top out! The bottom is just an addition, which collapses to a small number — once you know it, look for that number hiding as factors inside the top.
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Hint 2 of 2
Technique: never expand a product when you can cancel. Factor the denominator and hunt for those same factors among the terms upstairs.
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Approach: collapse the sum, then cancel matching factors
- First the easy part — the denominator is a sum, not a product: 1 + 2 + … + 8 = 36. (Pair 1+8, 2+7, 3+6, 4+5 = four 9's, or use 8·9/2.) And 36 = 6 · 6.
- Now cancel instead of multiply: the top already has a 6, and 2 · 3 builds a second 6 — both 6's wipe out the denominator.
- What survives on top: 1 · 4 · 5 · 7 · 8 = 4·5·7·8 = 1120.
- Why this transfers: a fraction of products is an invitation to cancel, never to brute-force; reduce before you ever multiply.
Another way — cancel using 36 = 4 · 9:
- Write 36 = 4 · 9. The top contains a 4 (cancel it) and a 9 living inside 3 · 6 = 18 = 9 · 2 (cancel the 9, leaving a 2 behind).
- After canceling the 4, the 3, and replacing the 6 with 2: top = 1 · 2 · 2 · 5 · 7 · 8 = 1120 — same answer, different factors canceled.
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