Problem 24 · 2020 Math Kangaroo
Stretch
Number Theory
divisibilityfactorizationperfect-square
Let N be the smallest positive number such that half of N is divisible by 2, one-third of N is divisible by 3, one-quarter of N is divisible by 4, one-fifth of N is divisible by 5, one-sixth of N is divisible by 6, one-eighth of N is divisible by 8, and one-ninth of N is divisible by 9. The square root of N is a number of how many digits?
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Answer: A — 3
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Hint 1 of 2
'Half of N divisible by 2' means N is a multiple of 4; turn each clue into a divisibility of N.
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Hint 2 of 2
Take the least common multiple of all those requirements, then count the digits of its square root.
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Approach: translate the clues, then take the LCM
- The clues say N is a multiple of 4, 9, 16, 25, 36, 64 and 81.
- Their least common multiple is N = 129600.
- √129600 = 360, which has 3 digits.
- The answer is 3, choice A.
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