Proof by contradiction Quiz

1. To prove by contradiction that $\sqrt5$ is irrational, which assumption do you make?

• A. $\sqrt5$ is irrational.
• B. $\sqrt5$ is an integer.
• C. $\sqrt5=\frac{a}{b}$ in lowest terms.
• D. $\sqrt5$ is prime.

2. If $a^2$ is divisible by 5, which can you conclude?

• A. $a$ is divisible by 5.
• B. $a=5$.
• C. $a^2=25$.
• D. $a$ is even.

3. In Euclid's proof with primes $2,3,5$, $N=2\times3\times5+1=31$. What does this show?

• A. 31 is divisible by 2.
• B. There are only three primes.
• C. 31 is composite.
• D. The list is incomplete, since 31 is a prime not in the list.

4. Which is a valid contradiction in a proof about a fraction $\frac{a}{b}$?

• A. $a$ and $b$ are integers.
• B. $a$ and $b$ are both even, though assumed to have no common factor.
• C. $a>0$.
• D. $b\neq0$.

5. Which is the negation of 'there is an integer solution to $x^2=2y^2$ with $y\neq0$'?

• A. There is a rational solution.
• B. There are no integer solutions with $y\neq0$.
• C. There are exactly two integer solutions.
• D. $x$ and $y$ are both zero.