Convert the decimal number 1071 to base 12.

Write the decimal number 1071 as a product of its prime factors.

Using your answers to part (a) and (b), prove that there is only one possible value for \(b\) and state this value.

Hence state the value of \(a\).

EITHER

using a list of relevant powers of 12: 1, 12, 144     (M1)

\(1071 = 7 \times {12^2} + 5 \times {12^1} + 3 \times {12^0}\)     (A1)

OR

attempted repeated division by 12     (M1)

\(1071 \div 12 = 89{\text{rem}}3;{\text{ }}89 \div 12 = 7{\text{rem}}5\)     (A1)

THEN

\(1071 = {753_{12}}\)     A1

[3 marks]

\(1071 = 3 \times 3 \times 7 \times 17\)     A1

[1 mark]

in base \(b\) \(a060\) ends in a zero and so \(b\) is a factor of 1071     R1

from part (a) \(b < 12\) as \(a060\) has four digits and so the possibilities are

\(b = 3,{\text{ }}b = 7\) or \(b = 9\)     R1

stating valid reasons to exclude both \(b = 3\) eg, there is a digit of 6

and \(b = 9\) eg, \(1071 = {(1420)_9}\)     R1

\(b = 7\)     A1

Note:     The A mark is independent of the R marks.

[4 marks]

\(1071 = {(3060)_7} \Rightarrow a = 3\)     A1

[1 mark]