Problem:
Each piece of candy in a shop costs a whole number of cents. Casper has exactly enough money to buy either pieces of red candy, pieces of green candy, pieces of blue candy, or pieces of purple candy. A piece of purple candy costs cents. What is the least possible value of ?
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Solution:
Because Casper has exactly enough money to buy pieces of red candy, the amount of money he has must be a multiple of cents. Similarly, it must be a multiple of both cents and cents. Furthermore, this amount of money will buy a whole number of purple candies that cost cents each, so the amount of money must also be a multiple of . The least common multiple of , , and is . Therefore the number of purple candies that Casper can buy, , must be a multiple of . Thus the least possible value of is . In this case the red candies cost cents each, the green candies cost cents each, and the blue candies cost cents each.
The problems on this page are the property of the MAA's American Mathematics Competitions