2020 CIME I Problems/Problem 2

Problem 2

At the local Blast Store, there are sufficiently many items with a price of $$n.99$ for each nonnegative integer $n$ . A sales tax of $7.5\%$ is applied on all items. If the total cost of a purchase, after tax, is an integer number of cents, find the minimum possible number of items in the purchase.

Solution

If $k$ items were purchased, the total price before the sales tax is $100N-k$ cents for some positive integer $N$ . After the sales tax of $7.5\%$ is applied, the price before tax is multiplied by $\frac{43}{40}$ . Thus we need $\frac{43}{40}(100N-k)$ to be an integer. This implies that $\frac{4300N-43k}{40}$ is an integer, so $4300N-43k$ is a multiple of $40$ . This condition implies that $100N-k$ is a multiple of $40$ because $43$ and $40$ are relatively prime. If $N$ is odd, the least possible value of $k$ is $20$ ; if $k$ is even, the least possible value is $40$ . The smaller of these is obviously $\boxed{20}$ .

2020 CIME I (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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2020 CIME I Problems/Problem 2

Problem 2

Solution

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