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

Given that is odd, use proof by contradiction to show, using algebra, that is even.

Given that is odd, use proof by contradiction to show, using algebra, that is odd.

A student is attempting to answer the following exam question:

“Prove by contradiction that  is an irrational number. You may use without proof the fact that if a number is even, then must also be even.”

The student’s proof is as follows:

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

Find the error and write down the correct line of the proof.

Line 4 of the proof is missing.

Complete this line of the proof.

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

Given that is odd, use proof by contradiction to show, using algebra, that is odd.

"There are an infinite number of positive multiples of 10."

A proof by contradiction starts as follows:

Write the statements needed to complete the proof.

A student attempts to answer the following question:

The student starts the proof with:

Complete the proof.

Given that and are integers such that

is even

use algebra to prove by contradiction that at least one of or is even.

Prove by contradiction that there are no positive integers and such that

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

Prove by contradiction that  is an irrational number. 

You may use without proof the fact that if is a multiple of 11, then is a multiple of 11.

Below is an attempt at a proof by contradiction to show that there is no largest multiple of 7.

Both line 2 and line 6 are incomplete.

Complete these lines of the proof.

Given that  is odd where is a positive integer, use proof by contradiction to show, using algebra, that is odd.

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

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

Below is a proof by contradiction that  is irrational.

Lines 4 is missing.

Complete this line of the proof.

Without solving the equation directly, use algebra to prove by contradiction that the solutions to the equation 

cannot be written in the form  where  and  are both odd integers.

A composite number, , has the following properties:

• It is a positive integer greater than 1

• It is not a prime number

• It has at least two prime factors

• It can be written as a product of its prime factors, , , ..., :

where .

Prove by contradiction that any composite number, , must have at least one prime factor that is less than or equal to .

Prove by contradiction that , where  is a prime number, is an irrational number. 

You may use without proof the facts that:

• Any positive integer may be written uniquely as a product of the powers of its prime factors,

• The powers of the prime factors of a square number are even, for example