Problem:
The base-ten representation for is , where , and denote digits that are not given. What is ?
Answer Choices:
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E.
Solution:
Because , , and all have a single factor of in their prime factorization, ends with s. Hence . To determine and , divisibility tests for and can be used. Because is divisible by , its digit sum, , must also be divisible by , which implies or . Similarly, because is divisible by , its alternating digit sum, , must also be divisible by . This implies that or . Combining these constraints results in only one solution in which and are digits, namely and . Hence .
The problems on this page are the property of the MAA's American Mathematics Competitions