Tourist Trouble
On the last day of his trip, a tourist from Paupa New Guinea wanted to buy a $7 vase from my shop. However, he ran out of Australian dollars and only had Paupa New Guinean coins--both circular and triangular--with him. The exchange rate on that day was that 7 circular coins were worth $6 and 7 triangular coins were worth $11.
Would he be able to pay for the vase without requiring change from me?
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1 solution
Each circular coin is worth 7 6 Australian dollars and each triangular coin is worth 7 1 1 Australian dollars. So to be able to pay without change being made, there must exist non-negative integers m , n such that
7 6 m + 7 1 1 n = 7 ⟹ 6 m + 1 1 n = 4 9 .
But by the Chicken McNugget Theorem we know that the greatest integer that cannot be written in this form is m n − m − n = 6 × 1 1 − 6 − 1 1 = 4 9 , and so the answer is No .