Find the general solution to the Diophantine equation $3x + 5y = 7$.

Find the values of $x$ and $y$ satisfying the equation for which $x$ has the smallest positive integer value greater than $50$.

by any method including trial and error or the Euclidean algorithm, a specific solution is, for example, $x = 4,{\text{ }}y =  - 1$     (A1)(A1)

$3(4) + 5( - 1) = 7\;\;\;$(equation i)

$3x + 5y = 7\;\;\;$(equation ii)

equation ii – equation i: $3(x - 4) + 5(y + 1) = 0$

$\frac{{4 - x}}{5} = \frac{{y + 1}}{3} = N$     (M1)

$x = 4 - 5N$     A1

$y = 3N - 1$     A1

smallest positive integer $> 50$ occurs when $N =  - 10$     (M1)

$x = 54,{\text{ }}y =  - 31$     A1

This question was well answered in general although a minority appeared not to realise that Diophantine means a solution in integers. It was expected that candidates would find a particular solution by inspection but some took a little longer by going via the Euclidean Algorithm.

This question was well answered in general although a minority appeared not to realise that Diophantine means a solution in integers. It was expected that candidates would find a particular solution by inspection but some took a little longer by going via the Euclidean Algorithm.