Problem 7 · 1986 AJHSME
Medium
Number Theory
bound-square-roots
How many whole numbers are between √8 and √80?
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Answer: B — 6.
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Hint 1 of 2
You don't need the messy decimal values of √8 and √80 — you only need to know which two *whole* numbers each one is trapped between. Which perfect squares sit just below and just above 8? And around 80?
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Hint 2 of 2
Once each square root is pinned between two integers, just list the whole numbers caught in between (carefully decide whether the endpoints count).
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Approach: trap each root between neighboring perfect squares
- √8 lives between √4 = 2 and √9 = 3, so 2 < √8 < 3. √80 lives between √64 = 8 and √81 = 9, so 8 < √80 < 9. You never compute a decimal — the perfect squares fence the roots in.
- Now count the whole numbers strictly between 2.something and 8.something: 3, 4, 5, 6, 7, 8 — that's 6.
- Why this works generally: to locate any √N, find the perfect squares just below and above N; that brackets the root without a calculator. Trap to watch: √8 ≈ 2.83, so 2 is *not* included, and √80 ≈ 8.94, so 8 *is*.
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