Problem:
The -digit number is divisible by . What is the remainder when this number is divided by
Answer Choices:
A.
B.
C.
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E.
Solution:
Notice that a number is divisible by if and only if the sum of its digits is also divisible by . The sum of the digits of the -digit number in the problem is:
Since is divisible by , must also be divisible by . Also, since is a digit, we know that intuitively . This means that can only be . Now we know that the -digit number in question is , and we want to find the remainder when we divide by . To solve this, simply use long division to see that:
Therefore, the remainder is . Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions