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What is the sum of all positive integer solutions less than or equal to \(20\) to the congruence \( 13(3x-2)\equiv 26\pmod 8?\)

 May 29, 2023
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Subtracting 26 from both sides of the congruence gives 13(3x - 2) = 0 (mod 8). Dividing both sides by 13 gives 3x - 2 = 0 (mod 8). Adding 2 to both sides gives 3x = 2 (mod 8). Dividing both sides by 3 gives x = 2/3 (mod 8).

The positive integer solutions to this congruence are 2/3, 10/3, 18/3, and 26/3. The sum of these solutions is 2 + 10 + 18 + 26 = 56.

Therefore, the answer is 56.

 May 29, 2023

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