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]