As n is a positive integer. is also positive. The right side can simply be referred to as m, where m is a positive even integer. (1 mark for recognising that is even)

since (x + 1) and (x + 2y +1) must multiply to make an even number, this can only be true when either both terms are even, or 1 term is odd, and the other term is even.

Scenario 1: Both terms are even.

x + 1 is even, therefore x is odd. Then x + 2y + 1 is even, so this scenario is possible. (1 mark)

Scenario 2: First term is odd and the second term is even.

x + 1 is odd, therefore X is even. Then x + 2y + 1 is odd. This is impossible as both x and y are even, thus x + 2y + 1 must be odd. This scenario is not possible as both terms are odd. (1 mark)

Scenario 3: the second term is odd and the first term is even.

If x +1 is even, x is odd. This will give the same answer as scenario 1 where x + 2y + 1 is even, so this scenario doesn’t exist. (1 mark)

Therefore x must be odd.

Award full marks for any alternative method that concludes x is odd.