Problem 25 · 2010 Math Kangaroo
Hard
Number Theory
divisibility
On each of 18 cards either a 4 or a 5 is written. The sum of the numbers on all the cards is divisible by 17. On how many cards is the number 4 written?
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Answer: B — 5
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Hint 1 of 2
Write the total in terms of how many cards show 4.
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Hint 2 of 2
Then make that total a multiple of 17.
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Approach: set up the sum and use divisibility
- If x cards show 4, the rest (18−x) show 5, so the sum is 4x + 5(18−x) = 90 − x.
- For 90 − x to be a multiple of 17 with 0 ≤ x ≤ 18, we need x = 5.
- So 5 cards show the number 4.
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