Problem 28 · 2009 Math Kangaroo
Stretch
Number Theory
factorizationperfect-square
What is the smallest whole number n for which the expression (2²−1)·(3²−1)·(4²−1)·…·(n²−1) is a square number?
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Answer: B — 8
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Hint 1 of 2
Factor each k²−1 as (k−1)(k+1) and see what stays unpaired.
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Hint 2 of 2
Test the running product for being a perfect square as n grows.
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Approach: track the running product and test for a square
- Each factor is k²−1 = (k−1)(k+1); the product up to n is (n−1)!·(n+1)!/2.
- Checking n = 2,3,… the product first becomes a perfect square at n = 8 (its value is 25401600 = 5040²).
- So the smallest n is 8.
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