(i)     Use the Euclidean algorithm to find gcd(\(6750\), \(144\)) .

(ii)     Express your answer in the form \(6750r + 144s\) where r , \(s \in \mathbb{Z}\) .

Consider the base \(15\) number CBA, where A, B, C represent respectively the digits ten, eleven, twelve.

  (i)     Write this number in base \(10\).

  (ii)     Hence express this number as a product of prime factors, writing the factors in base 4.

(i)    \(6750 = 46 \times 144 + 126\)     M1A1

\(144 = 1 \times 126 + 18\)     A1

\(126 = 7 \times 18\)

\(\gcd(6750,144) = 18\)     A1 N0

(ii)     \(18 = 144 - 1 \times 126\)     (M1)

\( = 144 - (6750 - 46 \times 144)\)

\( = 47 \times 144 + ( - 1) \times 6750\)     A1

[6 marks]

(i)     \(n = 10 + 11 \times 15 + 12 \times {15^2}\)     (M1)(A1)

\( = 2875\)     A1

(ii)     \(2875 = {5^3} \times 23\)    A1

\( = 11 \times 11 \times 11 \times 113\) in base \(4\)     A1A1

Note: A1 for \(11 \times 11 \times 11\) ,  A1 for \(113\) .

[6 marks]

Most candidates were able to use the Euclidean algorithm to find a gcd and to express the answer in a different form. All but the weakest candidates were able to make a meaningful start to this question.

In part (b) many correct answers were seen and a majority of students were able to use the work they had studied on number bases correctly. All but the weakest candidates were able to make a meaningful start to this question.