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imc

number theory intermediate

Problem

Let denote the smallest positive integer that is divisible by both and and whose base- representation consists of only 's and 's, with at least one of each. What are the last four digits of
(A)
(B)
(C)
(D)
Solution
For a number to be divisible by the last two digits have to be divisible by That means the last two digits of this integer must be For a number to be divisible by the sum of all the digits must be divisible by The only way to make this happen is with nine 's. However, we also need one The smallest integer that meets all these conditions is . The last four digits are
Final answer
C