Problem 7 · 2020 Math Kangaroo
Medium
Number Theory
divisibility
Which of the following expressions is divisible by 3 for every integer n?
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Answer: E — \(n^3 - n\)
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Hint 1 of 2
Try a few values of n in each option and watch the remainder mod 3.
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Hint 2 of 2
A product of three consecutive integers is always a multiple of 3.
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Approach: recognize three consecutive integers
- n³ − n = (n−1)·n·(n+1) is a product of three consecutive integers.
- Among any three consecutive integers one is a multiple of 3.
- So option E is divisible by 3 for every integer n (the others fail for some n).
Mark:
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