Problem 19 · 2014 Math Kangaroo
Hard
Number Theory
divisibilitycasework
Among 10 different positive whole numbers, exactly 5 are divisible by 5 and exactly 7 are divisible by 7. Let M be the biggest of these numbers. What is the smallest possible value of M?
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Answer: E — another value
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Hint 1 of 2
Multiples of 5 and multiples of 7 must fit inside just 10 numbers — they have to overlap.
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Hint 2 of 2
Overlap means multiples of 35; how few of those can you get away with, and how small can they be?
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Approach: count the forced overlap, then keep everything small
- Five multiples of 5 plus seven multiples of 7 is 12 'slots' but only 10 numbers, so at least 2 must be multiples of 35.
- The two smallest multiples of 35 are 35 and 70, so the largest number M is at least 70.
- Choosing 5,10,15,35,70 (the 5's) and 7,14,21,28,35,42,70 worth of 7's gives a valid set of 10 with M = 70.
- 70 is none of 105, 77, 75, 63, so the answer is (E) another value.
Mark:
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