- #1

Math100

- 762

- 206

- Homework Statement
- Find all solutions of the linear congruence ## 3x-7y\equiv 11\pmod {13} ##.

- Relevant Equations
- None.

Consider the linear congruence ## 3x-7y\equiv 11\pmod {13} ##.

Then ## 3x\equiv 11+7y\pmod {13} ##.

Note that ## gcd(3, 13)=1 ## and ## 1\mid 11 ##.

This means ## y\in\ {0, 1,..., 12} ##.

And now I'm stuck. The book's answer shows ## x\equiv 11+t\pmod {13}, y\equiv 5+6t\pmod {13} ##. But based on my previous work shown above, how should I get to the correct answer for this problem?

Then ## 3x\equiv 11+7y\pmod {13} ##.

Note that ## gcd(3, 13)=1 ## and ## 1\mid 11 ##.

This means ## y\in\ {0, 1,..., 12} ##.

And now I'm stuck. The book's answer shows ## x\equiv 11+t\pmod {13}, y\equiv 5+6t\pmod {13} ##. But based on my previous work shown above, how should I get to the correct answer for this problem?