Asif notices that \(24 ^ { 2 } = 576\) and \(2 + 4 = 6\) gives the last digit of 576
He checks two more examples:
$$\begin{array} { l c }
27 ^ { 2 } = 729 & 29 ^ { 2 } = 841
2 + 7 = 9 & 2 + 9 = 11
\text { Last digit } 9 & \text { Last digit } 1
\end{array}$$
Asif concludes that he can find the last digit of any square number greater than 100 by adding the digits of the number being squared.
Give a counter example to show that Asif's conclusion is not correct.
6
Claire tells Asif that he should look only at the last digit of the number being squared.
$$\begin{array} { c c }
27 ^ { 2 } = 729 & 24 ^ { 2 } = 576
7 ^ { 2 } = 49 & 4 ^ { 2 } = 16
\text { Last digit } 9 & \text { Last digit } 6
\end{array}$$
Using Claire's method determine the last digit of \(23456789 { } ^ { 2 }\) [0pt]
[1 mark]
6
Given Claire's method is correct, use proof by exhaustion to show that no square number has a last digit of 8