Proof by Contradiction (Edexcel International A Level (IAL) Maths: Pure 4): Exam Questions

Prove by contradiction that if $x^2$ is odd, then $x$ must be odd.

When a number is rational, it can be written in the form $\frac{a}{b}$.

(i) Write down the condition that $a$ and $b$ must satisfy.

(ii) Write down a further condition on $b$.

Two numbers can be written in the form  $\frac{p}{q}$ and $\frac{r}{s}$ such that  $p,q,r\mathrm{and}s$ meet the necessary conditions so that the two numbers are rational.

Prove that the product of these numbers is also rational.

Prove by contradiction that there are an infinite number of even numbers.

A student is attempting to answer the following exam question:

“Prove by contradiction that $\sqrt{2}$ is an irrational number. You may use without proof the fact that if a number $n^2$ is even, then n must also be even.”

The student’s proof proceeds as follows:

There is an error within the first three lines of the proof.

State what the error is and write the correct line down.

Line 4 of the proof is missing.

Write down the missing line of the proof.

(i) How many distinct factors does a prime number have?

(ii) What can you say about the number of distinct factors a square number has?

(iii) If N is the square of a prime number, then excluding N itself, write down, in terms of N, the largest factor of N.

A composite number can be written uniquely as the product of its prime factors.i.e., any composite number N can be written uniquely as $N=p_1\times p_2\times p_3\times ...$, where $p_1,p_2,p_3,...$ are the prime factors of N.

Show that a composite number N may be written in the form $N=pq$, where $q$ is an integer and $p$ is a prime factor of N.

By expressing $q$ in terms of the prime factors of N, be sure to explain why $q$ must be an integer.

Prove by contradiction that a triangle cannot have more than one obtuse angle.

Prove by contradiction that if $x^3$ is odd, then $x$ must be odd.

Prove that the product of two rational numbers is rational.

Prove by contradiction that there are an infinite number of powers of 2.

Prove by contradiction that $\sqrt{11}$ is an irrational number.  You may use without proof the fact that if $n^2$ is a multiple of 11, then $n$ is a multiple of 11.

Below is a proof by contradiction that there is no largest multiple of 7.

The proof contains two omissions in its argument.

Identify both omissions and correct them.

If a positive integer greater than 1 is not a prime number, then it is called a composite number.  Prove by contradiction that any composite integer N has a prime factor less than or equal to $\sqrt{N}$.

Prove by contradiction that if $x^n$ is odd, where $n\ge 2$ is a positive integer, then $x$ must be odd.

Prove that the difference between two rational numbers is rational.

Prove by contradiction that there are an infinite number of prime numbers.

Prove by contradiction that $\sqrt{k}$, where $k$ is a prime number, is an irrational number.  You may use without proof the fact that any positive integer may be written uniquely as a product of its prime factors.

Below is a proof by contradiction that ${\log }_{2}7$ is irrational.

The proof contains one mathematical error and one logical error.

Identify both errors and correct them.

Prove by contradiction that the solutions to the equation  $3x^2+10x-8=0$  cannot be written in the form $\frac{a}{b}$  where $a$ and $b$ are both odd integers.

Prove by contradiction that, if $p,q,r\mathrm{and}s$ are rational numbers and $c$ is a positive non-square integer, then

$p+q\sqrt{c}=r+s\sqrt{c}$

implies that $p=r$ and $q=s$.  You may use without proof the fact that for any positive non-square integer $n$, $\sqrt{n}$ is irrational.