A number written in base 5 is 4303. Find this as a number written in base 10.

1000 is a number written in base 10. Find this as a number written in base 7.

4303₅ = 4 × 5³ + 3 × 5² + 0 × 5¹ + 3 × 5⁰        (M1)
= 500 + 75 + 3
= 578        A1

[2 marks]

METHOD 1

$1000 = {a_0} + 7{a_1} + 49{a_2} + 343{a_3}$      (M1)

(since $343{a_3} < 1000$)$\Rightarrow {a_3} = 2$     (A1)

$1000 = {a_0} + 7{a_1} + 49{a_2} + 686$

${a_0} + 7{a_1} + 49{a_2} = 314 \Rightarrow {a_2} = 6$     (A1)

${a_0} + 7{a_1} = 20 \Rightarrow {a_1} = 2,\,\,{a_0} = 6$     (A1)

$\Rightarrow {1000_{10}} = {2626_7}$      A1

METHOD 2

1000 = 7 × 142 + 6      (M1)

142 = 7 × 20 + 2     (A1)

20 = 7 × 2 + 6     (A1)

2 = 7 × 0 + 2     (A1)

⇒ (1000)₁₀ = 2626₇      A1

[5 marks]