9.10 Coursework Sheet 9

Hand in your solutions in the appropriate way by Monday, 11 December 2023, 19:00.

Exercise 1

Find all the solutions to the following congruence:

\[x^{26}-x^{19}+12x^6+4\equiv 0\hbox{ mod }7.\]

Exercise 2

Show that \(a^{49}-a\) is divisible by \(70\) for each integer \(a\).

Exercise 3*

Prove that for any odd prime \(p\), \(2^{p-1}+2(p-3)!\) is divisible by \(p\). [Hint: use Fermat’s and Wilson’s Theorems]

Exercise 4

Solve the congruence \(x^2\equiv1011 \hbox{ mod }2063\). (You may assume 2063 is a prime).

Exercise 5

Exam question 13, 2007/8 List each of the units in \({\mathbb Z}_{20}\) giving the multiplicative inverse in each case.

Exercise 6

Compute \(\phi(n)\) in each of the following cases: i) \(n=24\), ii) \(n=600\), iii) \(n=1111\).

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