OCR
Further Additional Pure
2018
March
— Question 4
Exam Board
OCR
Module
Further Additional Pure (Further Additional Pure)
Year
2018
Session
March
Topic
Number Theory
4
(a) Find all the quadratic residues modulo 11.
(b) Prove that the equation \(y ^ { 5 } = x ^ { 2 } + 2017\) has no solution in integers \(x\) and \(y\).
In this question you must show detailed reasoning.
The numbers \(M\) and \(N\) are given by
$$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$
Prove that \(M\) is divisible by \(N\).