aandbare consecutive odd numbers, wherea>b.

Prove thata^{3}-b^{3}is 2 more than a multiple of 24.

Fill in the gaps in the following proof.

ais odd, soa= 2n+1 for some integern.

Then in terms ofn,b=..............

soa^{3}-b^{3}= (2n+ 1)^{3}- ...............

Expanding the brackets and simplifying,

a^{3}-b^{3}= ............+ 2which is 2 more than a multiple of 24, as required.

is b = 2n +1 <a ?