Problem 28 · 2016 Math Kangaroo
Stretch
Number Theory
divisibilityprimes
Anna chooses a positive whole number n and writes down the sum of all positive whole numbers from 1 to n. A prime number p divides this sum but none of the summands. Which of the following numbers is a possible value of n + p?
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Answer: A — 217
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Hint 1 of 3
The sum \(1+2+\cdots+n\) equals \(\tfrac{n(n+1)}{2}\).
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Hint 2 of 3
A prime that divides the sum but none of the summands \(1,\dots,n\) must be larger than \(n\).
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Hint 3 of 3
The only way such a prime appears in \(\tfrac{n(n+1)}{2}\) is as \(n+1\) itself.
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Approach: the special prime must equal n+1
- The sum is \(\tfrac{n(n+1)}{2}\); if prime \(p\) divides it but no summand \(1,\dots,n\), then \(p > n\).
- A prime bigger than \(n\) can only come from the factor \(n+1\), so \(p = n+1\) and \(n+p = 2n+1\) is odd.
- Test the odd options: \(217 = 2(108)+1\) needs \(n=108\), \(p=109\), and 109 is prime, while the others give composite \(n+1\).
- For \(n=108\), \(p=109\) divides \(\tfrac{108\cdot109}{2}\) but no summand, so 217 works (A).
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