Emmy says to Max, βI ordered math club sweatshirts today.β Max asks, βHow much did each shirt cost?β Emmy responds, βIβll give you a hint. The total cost was , where and are digits and .β After a pause, Max says, βThat was a good price.β What is
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The total cost is , with digits and , . In cents, let represent the total cost of the sweatshirts which is the -digit integer .
Since there are sweatshirts, the total in cents must be divisible by . Therefore is divisible by and by . Now, we can solve it in ways, one of which uses modular arithmetic:
Solution 1: (Modular Arithmetic)
We know that the total price is divisible by and . The last two digits are , so is divisible by , so
Since the price is also divisible by , the sum of the digits is
From ,
Testing with and a nonzero digit, the only pair that also satisfies (1) is . Thus, the total cost is , and
Solution 2: (Divisibility Rules):
The divisibility rule for says that the sum of digits must be divisible by , so for some integer
As and are divisible by , we know that must also be divisible by . Furthermore, as is divisible by , it is even, and hence is even as well.
Since is even, divisible by , and is a nonzero digit, . To find , we note that since , our possibilities for are . However, since is divisible by , divisibility rules say that is divisible by , implying . Therefore,
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