2020 CIME I Problems/Problem 2
At the local Blast Store, there are sufficiently many items with a price of for each nonnegative integer . A sales tax of 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.
If items were purchased, the total price before the sales tax is cents for some positive integer . If the sales tax of is applied, the price before tax is multiplied by . Thus we need to be an integer. This implies that is an integer, so is a multiple of . This condition implies that is a multiple of because and are relatively prime. If is odd, the least possible value of is ; if is even, the least possible value is . The smaller of these is obviously .
https://www.youtube.com/watch?v=gdOnSyFewDU ~Shreyas S
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