Problem 28 · 2014 Math Kangaroo
Stretch
Number Theory
divisibilitycasework
Several different positive whole numbers are written on a blackboard. Exactly two of these numbers are divisible by 2, and exactly 13 of these numbers are divisible by 13. The biggest number on the board is M. What is the smallest value that M can have?
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Answer: C — 273
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Hint 1 of 2
You need 13 multiples of 13, but only 2 of all the numbers may be even.
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Hint 2 of 2
Use odd multiples of 13 as much as possible; how many odd multiples of 13 do you need, and how big is the last one?
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Approach: use odd multiples of 13 to respect the 'only 2 even' limit
- Thirteen of the numbers are multiples of 13, but at most 2 numbers overall may be even.
- An even multiple of 13 is also divisible by 2, so among the thirteen multiples at most 2 can be even, meaning at least 11 must be odd multiples of 13.
- The odd multiples of 13 are 13·1, 13·3, 13·5, …; the 11th of these is 13 × 21 = 273.
- So the largest number M is at least 273.
Mark:
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