4. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
- (a) Use the Euclidean algorithm to find the highest common factor \(h\) of 416 and 72
(b) Hence determine integers \(a\) and \(b\) such that
$$416 a + 72 b = h$$
(c) Determine the value \(c\) in the set \(\{ 0,1,2 \ldots , 415 \}\) such that
$$23 \times 72 \equiv c ( \bmod 416 )$$ - Evaluate \(5 ^ { 10 } ( \bmod 13 )\) giving your answer as the smallest positive integer solution.