Problem 12 · 2025 Math Kangaroo
Medium
Number Theory
divisibilityfactorization
Among 10 different given positive integers, exactly five are divisible by 5 and exactly seven are divisible by 7. Let M be the largest of these numbers. What is the smallest possible value of M?
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Answer: E — a different value
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Hint 1 of 2
At least how many numbers must be divisible by both 5 and 7?
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Hint 2 of 2
Inclusion–exclusion forces ≥2 multiples of 35; the second-smallest is 70, so M is at least 70.
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Approach: inclusion–exclusion, then minimise the maximum
- 5 + 7 − (multiples of 35) ≤ 10, so at least 2 numbers are multiples of 35.
- The two smallest multiples of 35 are 35 and 70, so 70 must appear and is the largest.
- M = 70, which is not among A–D, so the answer is a different value.
Mark:
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