diophantine equation

[logistic_guy]

Show that there are no solutions to the following linear Diophantine equation:

\(\displaystyle 15x + 12y = 13\)

 

[logistic_guy]

First step, find the \(\displaystyle \text{GCD}\) of \(\displaystyle 15\) and \(\displaystyle 12\).

\(\displaystyle 15 = 3 \times 5\)
\(\displaystyle 12 = 2^2 \times 3\)

So, the \(\displaystyle \text{GCD}(15,12) = 3\).

Since \(\displaystyle 3\) is not a divisor of \(\displaystyle 13\), there are no solutions.