1.15 Proof by induction, contradiction and counterexample Quiz

1. Which is the correct basis for proving $7^{n} - 1$ is divisible by 6 for $n \in \mathbb{Z}^{+}$?

• A. $7^{1} - 1 = 6$, which is divisible by 6
• B. $7^{0} - 1 = 0$, which is divisible by 6
• C. Assume $7^{k} - 1 = 6m$
• D. $7^{2} - 1 = 48$, which is divisible by 6

2. In proving $\sqrt{2}$ is irrational, what is assumed at the start?

• A. $\sqrt{2} = \frac{p}{q}$ with $p, q$ integers having no common factor
• B. $\sqrt{2}$ is irrational
• C. $p$ and $q$ are both even
• D. $\sqrt{2}$ is an integer

3. Which value of $n$ is a counterexample to '$2^{n} + 1$ is prime for all $n \in \mathbb{Z}^{+}$'?

• A. $3$
• B. $1$
• C. $2$
• D. $4$

4. In Euclid's proof, what can we say about N = p₁p₂…pₙ + 1?

• A. It has a prime factor not in the list
• B. It is always prime
• C. It is divisible by p₁
• D. It is even

5. Which statement is true?

• A. The sum of a rational and an irrational number is irrational
• B. The sum of two irrational numbers is irrational
• C. The product of two irrational numbers is irrational
• D. The square of an irrational number is irrational