Disproving a statement with a counterexample Revision Card

AI generatedAlgebraKS3 / GCSE

One valid counterexample is enough.

State that the variable represents an integer when proving integer properties.
Use algebra to cover every permitted value. Factor out the divisor or 2 when helpful.
One counterexample disproves a universal statement; several examples do not prove it.

Example 1

Disprove: “ is prime for every positive integer .”

One valid counterexample is enough.

This has factors 2 and 3, both greater than 1.

For

, the result is 6, which is not prime. The statement is false.

Example 2

Disprove: “ is prime for every positive integer .”

One valid counterexample is enough.

This has factors 3 and 4, both greater than 1.

For

, the result is 12, which is not prime. The statement is false.

Your turn

Question 1

Disprove: “ is prime for every positive integer .”

Check answer

One valid counterexample is enough.

This has factors 4 and 5, both greater than 1.

For

, the result is 20, which is not prime. The statement is false.

Question 2

Disprove: “ is prime for every positive integer .”

Check answer

One valid counterexample is enough.

This has factors 5 and 6, both greater than 1.

For

, the result is 30, which is not prime. The statement is false.

Question 3

Disprove: “ is prime for every positive integer .”

Check answer

One valid counterexample is enough.

This has factors 6 and 7, both greater than 1.

For

, the result is 42, which is not prime. The statement is false.

Question 4

Disprove: “ is prime for every positive integer .”

Check answer

One valid counterexample is enough.

This has factors 7 and 8, both greater than 1.

For

, the result is 56, which is not prime. The statement is false.

Question 5

Disprove: “ is prime for every positive integer .”

Check answer

One valid counterexample is enough.

This has factors 8 and 9, both greater than 1.

For

, the result is 72, which is not prime. The statement is false.