Proof by contradictionAQA A-Level Maths: Mind map

Exam questions on Proof by contradiction

1. A student wishes to prove by contradiction that $\sqrt3$ is irrational.
   Given that $p^2=3q^2$ and $p$ is a multiple of $3$, complete the proof that $\sqrt3$ is irrational.2 marks
2. Claim: there are no positive integers $a$ and $b$ such that $a^2-b^2=1$.
   Complete the proof that no such integers exist.2 marks
3. Euclid's proof that there are infinitely many prime numbers begins by assuming that there are only finitely many, say $p_1,p_2,\ldots,p_n$, and then considers the number $N=p_1p_2\cdots p_n+1$.
   Show that $N$ leaves remainder $1$ when divided by each of $p_1,p_2,\ldots,p_n$, and state what this means for the divisibility of $N$.3 marks