In base \(5\) an integer is written \(1031\). Express this integer in base \(10\).

Given that \(N = 365,{\text{ }}r = 10\) and \(s = 7\) find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }} \ldots \).

(i)     Given that \(N = 899,{\text{ }}r = 10\) and \(s = 12\) find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2},{\text{ }} \ldots \)

(ii)     Hence write down the integer in base \(12\), which is equivalent to \(899\) in base \(10\).

Show that \(121\) is always a square number in any base greater than \(2\).

\(1031 = 1 \times {5^3} + 0 \times {5^2} + 3 \times 5 + 1\)     M1

\( = 125 + 0 + 15 + 1\)

\( = 141\)     A1

\(365 = 1 \times {7^3} + 0 \times {7^2} + 3 \times 7 + 1\)     M1

\( \Rightarrow {a_0} = 1,{\text{ }}{a_1} = 3,{\text{ }}{a_2} = 0,{\text{ }}{a_3} = 1\)     A1

(i)     \(899 = 6 \times {12^2} + 2 \times 12 + 11\)     M1

\( \Rightarrow {a_0} = 11,{\text{ }}{a_1} = 2,{\text{ }}{a_2} = 6\)     A1

(ii)     \({(899)_{10}} = {(62B)_{12}}\)

(where \(B\) represents the digit in base \(12\) given by \({a_0} = 11\))     A1

Note: Accept any letter in place of \(B\) provided it is defined

\(121\) in base \(r\) is \(1 + 2r + {r^2}\)     M1A1

\( = {(r + 1)^2}\)     A1

which is a square for all \(r\)     AG

This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than \(2\).

This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than 2.

This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than 2.

This question was very well answered in general, although some candidates failed to see that \(121 = {11^2}\) in all number bases greater than 2.