Problem 10 · 1994 AJHSME
Hard
Number Theory
divisors
For how many positive integer values of N is the expression 36N + 2 an integer?
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Answer: A — 7.
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Hint 1 of 2
A fraction 36/(something) is a whole number exactly when that 'something' divides 36 evenly. So forget N for a second — ask which numbers go into 36.
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Hint 2 of 2
Don't list every divisor and stop there: N has to be a positive integer (N ≥ 1), so the bottom N+2 is at least 3. Toss out any divisor smaller than 3.
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Approach: count valid divisors of 36
- 36/(N+2) is an integer exactly when N+2 is a divisor of 36. All divisors of 36 (pair them up): 1·36, 2·18, 3·12, 4·9, 6·6 → 1, 2, 3, 4, 6, 9, 12, 18, 36.
- But N ≥ 1 forces N+2 ≥ 3, so the divisors 1 and 2 are off-limits (they'd need N = −1 or 0). Keep 3, 4, 6, 9, 12, 18, 36 — that's 7 values.
- Two ideas worth keeping: list divisors in PAIRS so you never miss one, and always re-check the hidden constraint (here N ≥ 1) before counting — the smaller divisors are the classic trap that turns the right list into a wrong total.
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