Proof by contradiction Quiz

1. In a proof by contradiction of a statement $S$, what is the first assumption?

• A. $S$ is true
• B. $S$ is false
• C. $S$ has a counterexample only if it is false
• D. $S$ is obvious

2. To prove $\sqrt5$ is irrational, which assumption begins the proof?

• A. $\sqrt5$ is an integer
• B. $\sqrt5=\frac{p}{q}$ with $p$, $q$ both multiples of $5$
• C. $\sqrt5=\frac{p}{q}$ with $p$, $q$ integers having no common factor
• D. $\sqrt5$ is irrational

3. From $p^2=5q^2$ with $p$, $q$ integers, what follows?

• A. $p$ is a multiple of $5$
• B. $p$ is even
• C. $q=1$
• D. $p=5$

4. In Euclid's proof, what remainder does $N=p_1p_2\cdots p_n+1$ leave on division by $p_1$?

• A. $0$
• B. $1$
• C. $p_1$
• D. $n$

5. Assume $3+2\sqrt2=\frac{a}{b}$. Which expression for $\sqrt2$ follows?

• A. $\frac{a-3}{2}$
• B. $\frac{a}{2b}-3$
• C. $\frac{a-3b}{b}$
• D. $\frac{a-3b}{2b}$