Problem 11 · 2011 Math Kangaroo
Medium
Number Theory
sum-constraintcareful-counting
Johannes has only 5 Cent coins and 10 Cent coins in his pocket. Altogether he has 13 coins. Which of the following amounts cannot be the total of his coins?
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Answer: B — 60 c
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Hint 1 of 2
With 13 coins, swapping a 5 c coin for a 10 c coin changes the total by a fixed step.
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Hint 2 of 2
Find the smallest and largest possible totals, and the step between reachable totals.
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Approach: range and step of the total
- All 5 c gives 65 c; all 10 c gives 130 c; each 5→10 swap adds 5 c, so totals are 65, 70, 75, …, 130.
- 80, 70, 115 and 125 are all multiples of 5 in that range, but 60 c is below 65 c, so it cannot be the total, answer B.
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