Problem 11 · 1988 AJHSME
Hard
Number Theory
bound-by-perfect-squares
√164 is
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Answer: E — between 12 and 13.
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Hint 1 of 2
You don't need the exact value — just find the two perfect squares that 164 sits between. Which squares do you know that bracket it?
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Hint 2 of 2
Run up the squares you know: …, 12² = 144, 13² = 169. 164 lands between those two, so its root lands between 12 and 13.
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Approach: trap 164 between consecutive perfect squares
- Find the perfect squares just below and just above 164: 12² = 144 and 13² = 169. Since 144 < 164 < 169, taking square roots keeps the order: 12 < √164 < 13.
- So √164 lies between 12 and 13.
- Trap to avoid: choice 42 tempts anyone who thinks a square root makes a number bigger — it doesn't. √164 is far smaller than 164.
- Why this transfers: to locate any square root, sandwich the number between two perfect squares you've memorized. Square roots preserve order, so the root lands between those two whole numbers.
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