Problem 3 · 2009 Math Kangaroo
Easy
Number Theory
factorization
For how many positive whole numbers n is \(n^2+n\) a prime number?
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Answer: B — 1
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Hint 1 of 2
Factor n² + n before testing whether it can be prime.
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Hint 2 of 2
A prime has only the factors 1 and itself—so when can a product of two integers be prime?
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Approach: factor, then force one factor to be 1
- n² + n = n(n + 1), a product of two consecutive positive integers.
- For that product to be prime, one factor must be 1, so n = 1, giving 1·2 = 2 (prime).
- For every n ≥ 2 both factors exceed 1, so the product is composite. Exactly 1 value works.
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