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.