9.9 Coursework Sheet 8
Hand in your solutions in the appropriate way by Monday, 04 December 2023, 19:00.
Exercise 1
Solve the congruence \(x^2\equiv 34\mod{55}\). (HINT: You can convert this to two congruences modulo \(5\) and \(11\) respectively, each of which is easier to solve.)
Exercise 2
In each of the following cases decide which of \(\pm y\) has a square root mod \(p\), and find the square roots.
- \(y= 5\), \(p=7\),
- \(y=5\), \(p=19\),
- \(y=5\) \(p=23\).
Exercise 4
Explain why it is ‘obvious’ that \(8^{28}\) is a solution of the congruence \(64x\equiv1\mod 31\).
Exercise 5
You are given that \(11413\) is the product of the two primes \(101\) and \(113\). You are also given that two of the square roots of \(2601 \mod{11413}\) are \(\pm 51\). Using the fact that the other square roots \(\pm b\) satisfy the simultaneous equations \(b\equiv -51\mod{101}\) and \(b\equiv 51\mod{113}\) solve to find the roots \(\pm b\).