Problem 22 · 2008 AMC 8
Medium
Number Theory
range-of-integers
For how many positive integer values of n are both n3 and 3n three-digit whole numbers?
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Answer: A — 12.
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Hint 1 of 2
For n/3 to even be a whole number, n must be a multiple of 3 — so write n = 3x and both conditions become conditions on x.
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Hint 2 of 2
Two range conditions overlap; you only need the tighter (binding) one on each end, then count the integers in the survivor range.
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Approach: substitute n = 3x, then find the binding range
- Since n/3 must be a whole number, let n = 3x. Then n/3 = x and 3n = 9x, so we need both x and 9x to be three-digit: 100 ≤ x ≤ 999 and 100 ≤ 9x ≤ 999.
- The big number 9x is the squeeze: 9x ≤ 999 forces x ≤ 111, far tighter than x ≤ 999. The lower end is just x ≥ 100.
- So x runs 100, 101, …, 111 — that's 111 − 100 + 1 = 12 values.
- Why this transfers: when several inequalities pin a variable, keep only the strictest on each side; and counting integers from a to b inclusive is b − a + 1 (don't forget the +1).
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