AI generatedAlgebraGCSE Higher

Proving a statement about parity

Express the result as an even or odd integer.

  • 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

Prove that the sum of and is even for every integer .

Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.

Example 2

Prove that the sum of and is even for every integer .

Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.

Your turn

Question 1

Prove that the sum of and is even for every integer .

Check answer
Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.
Question 2

Prove that the sum of and is even for every integer .

Check answer
Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.
Question 3

Prove that the sum of and is even for every integer .

Check answer
Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.
Question 4

Prove that the sum of and is even for every integer .

Check answer
Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.
Question 5

Prove that the sum of and is even for every integer .

Check answer
Add the two expressions and collect like terms.Factor out 2. The bracket is an integer.
The sum is twice an integer, so it is even.
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