Problem 25 · 2018 AMC 8
Hard
Number Theory
perfect-cubeestimate-and-pick
How many perfect cubes lie between 28 + 1 and 218 + 1, inclusive?
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Answer: E — 58 cubes.
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Hint 1 of 2
Counting cubes is really counting their cube-roots: a cube n3 is in range exactly when n is. So the whole game is finding the smallest and largest allowed n. Look hard at 218 — that exponent 18 is begging to be split as 6×3.
Still stuck? Show hint 2 →
Hint 2 of 2
The technique: take cube roots of the boundaries to turn a range of cubes into a range of integers, then count integers with "last − first + 1." The "+1" on each bound (the +1's in 28+1 and 218+1) only nudges the endpoints — check whether each endpoint is itself a cube.
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Approach: bracket the cube-root bounds
- Upper end: 218 = (26)3 = 643, which is ≤ 218 + 1, so base 64 counts but base 65 (= 653) blows past the limit. Largest base: 64.
- Lower end: 28 + 1 = 257. Bracket it between cubes: 63 = 216 < 257 but 73 = 343 ≥ 257, so the smallest cube in range has base 7.
- Count the integer bases from 7 to 64 inclusive: 64 − 7 + 1 = 58.
- You'll see it again: "how many cubes (or squares) in a range" collapses to "how many integers between the cube roots" — and recognizing a power like 218 as a perfect cube is the move that makes the top bound exact instead of approximate.
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