Problem 13 · 2010 Math Kangaroo
Stretch
Number Theory
divisibilityratio
The integers x and y fulfill the condition 2x = 5y. Only one of the following numbers can be considered for x + y. Which?
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Answer: C — 2009
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Hint 1 of 2
From 2x = 5y, write x and y in terms of one parameter.
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Hint 2 of 2
Then x + y is always a multiple of the same number.
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Approach: parametrise the equation, then test divisibility
- 2x = 5y means x = 5k and y = 2k for an integer k.
- Then x + y = 7k, so the sum must be a multiple of 7.
- Of the choices, only 2009 = 7×287 is a multiple of 7, so 2009.
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