Use the Euclidean algorithm to find \(\gcd (162,{\text{ }}5982)\).

The relation \(R\) is defined on \({\mathbb{Z}^ + }\) by \(nRm\) if and only if \(\gcd (n,{\text{ }}m) = 2\).

(i)     By finding counterexamples show that \(R\) is neither reflexive nor transitive.

(ii)     Write down the set of solutions of \(nR6\).

\(5982 = 162 \times 36 + 150\)    M1A1

\(162 = 150 \times 1 + 12\)    A1

\(150 = 12 \times 12 + 6\)

\(12 = 6 \times 2 + 0 \Rightarrow \gcd \) is 6     A1

[4 marks]

(i)     for example, \(\gcd (4,{\text{ }}4) = 4\)     A1

\(4 \ne 2\)    R1

so \(R\) is not reflexive     AG

for example

\(\gcd (4,{\text{ }}2) = 2\) and \(\gcd (2,{\text{ }}8) = 2\)     M1A1

but \(\gcd (4,{\text{ }}8) = 4{\text{ }}( \ne 2)\)     R1

so \(R\) is not transitive     AG

(ii)     EITHER

even numbers     A1

not divisible by 6     A1

OR

\(\{ 2 + 6n:n \in \mathbb{N}\} {\text{ }} \cup \{ 4 + 6n:n \in \mathbb{N}\} \)   A1A1

OR

\(2,{\text{ }}4,{\text{ }}8,{\text{ }}10,{\text{ }} \ldots \)    A2

[7 marks]

This was a successful question for many students with many wholly correct answers seen. Part (a) was successfully answered by most candidates and those candidates usually had a reasonable understanding of how to complete part (b). A number were not fully successful in knowing how to explain their results.

This was a successful question for many students with many wholly correct answers seen. Part (a) was successfully answered by most candidates and those candidates usually had a reasonable understanding of how to complete part (b). A number were not fully successful in knowing how to explain their results.