Use the Euclidean algorithm to find \(\gcd (752,{\text{ }}352)\).

A farmer spends £8128 buying cows of two different breeds, A and B, for her farm. A cow of breed A costs £752 and a cow of breed B costs £352.

(i)     Set up a diophantine equation to show this.

(ii)     Using your working from part (a), find the general solution to this equation.

(iii)     Hence find the number of cows of each breed bought by the farmer.

752 = 2(352) + 48     M1

352 = 7(48) +16     A1

48 = 3(16)     A1

therefore \(\gcd (752,{\text{ }}352)\) is 16     R1

[4 marks]

(i)     let x be the number of cows of breed A

let y be the number of cows of breed B

\(752x + 352y = 8128\)     A1

(ii)     \(16|8128\) means there is a solution

\(16 = 352 - 7(48)\)     (M1)(A1)

\(16 = 352 - 7\left( {752 - 2(352)} \right)\)

\(16 = 15(352) - 7(752)\)     (A1)

\(8128 = 7620(352) - 3556(752)\)

\( \Rightarrow {x_0} = - 3556,{\text{ }}{y_0} = 7620\)     (A1)

\( \Rightarrow x = - 3556 + \left( {\frac{{352}}{{16}}} \right)t = - 3556 + 22t\)

\( \Rightarrow y = 7620 - \left( {\frac{{752}}{{16}}} \right)t = 7620 - 47t\)     M1A1A1

(iii)     for x, y to be \( \geqslant 0\), the only solution is t = 162     M1

\( \Rightarrow x = 8,{\text{ }}y = 6\)     A1

[10 marks]

Part (a) of this question was the most accessible on the paper and was completed correctly by the majority of candidates. It was pleasing to see that candidates were not put off by the question being set in context and most candidates were able to start part (b). However, a number made errors on the way, quite a number failed to give the general solution and it was only stronger candidates who were able to give a correct solution to part (b) (iii).

Part (a) of this question was the most accessible on the paper and was completed correctly by the majority of candidates. It was pleasing to see that candidates were not put off by the question being set in context and most candidates were able to start part (b). However, a number made errors on the way, quite a number failed to give the general solution and it was only stronger candidates who were able to give a correct solution to part (b) (iii).