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