AI generatedAlgebraGCSE Higher

Proving a divisibility statement algebraically

Factor out the required 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 is divisible by 4 for every integer .

Expand the square.Collect like terms. This is 4 multiplied by the integer .
It is divisible by 4.

Example 2

Prove that is divisible by 6 for every integer .

Expand the square.Collect like terms. This is 6 multiplied by the integer .
It is divisible by 6.

Your turn

Question 1

Prove that is divisible by 8 for every integer .

Check answer
Expand the square.Collect like terms. This is 8 multiplied by the integer .
It is divisible by 8.
Question 2

Prove that is divisible by 10 for every integer .

Check answer
Expand the square.Collect like terms. This is 10 multiplied by the integer .
It is divisible by 10.
Question 3

Prove that is divisible by 12 for every integer .

Check answer
Expand the square.Collect like terms. This is 12 multiplied by the integer .
It is divisible by 12.
Question 4

Prove that is divisible by 14 for every integer .

Check answer
Expand the square.Collect like terms. This is 14 multiplied by the integer .
It is divisible by 14.
Question 5

Prove that is divisible by 16 for every integer .

Check answer
Expand the square.Collect like terms. This is 16 multiplied by the integer .
It is divisible by 16.
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