# 2009 UNCO Math Contest II Problems/Problem 3

## Problem

An army of ants is organizing a march to the Obama inauguration. If they form columns of $10$ ants there are $8$ left over. If they form columns of $7, 11$ or $13$ ants there are $2$ left over. What is the smallest number of ants that could be in the army?

## Solution

Let the smallest number of ants in the army be x. x modulo 10 is 8, so the last digit must be 8. The result of x modulo 7, 11, and 13 is 2 and the LCM of 7, 11, and 13 is 1001, so we can consider x modulo 1001 to be 2. Any multiple of 1001 plus 2 satisfies this condition, so all that must be done is finding the first instance of this where the last digit is 8. 1001*6 is 6006 and addition of 2 yields 6008, which is the smallest number of ants that could be in the army.

## See also

 2009 UNCO Math Contest II (Problems • Answer Key • Resources) Preceded byProblem 2 Followed byProblem 4 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 All UNCO Math Contest Problems and Solutions
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