Mock AIME 6 2006-2007 Problems/Problem 1

Problem

Let $T$ be the sum of all positive integers of the form $2^r\cdot3^s$ , where $r$ and $s$ are nonnegative integers that do not exceed $4$ . Find the remainder when $T$ is divided by $1000$ .

Solution

We note that the required sum is equal to $(2^0+2^1+2^2+2^3+2^4)(3^0+3^1+3^2+3^3+3^4)=(31)(121)=3751$ , which is $\framebox{751}$ mod 1000. ~AbbyWong

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Mock AIME 6 2006-2007 Problems/Problem 1

Problem

Solution