Question

prove that if a and b are both odd positive integers , then a^3-b^3 is even but not devisible by 4

Answer 1

let a be written as 2q + 1 and b be written as 2p+1
now, a^3 - b^3 = (2q +1)^3 - (2p+1)^3
expand it and you'll land up with
(2q-2p)(4q^2 + 4p^2 + 4pq + 3 +4p + 4q + 2p + 2q)
take 2 common and you'll land up with something multiplied by two...

now expand the expression by opening the brackets, take 4 common you'll land up with something like this

4m + 2
since remainder isn't equal to 0, though acube - bcube is even, it isn't divisible by 4;)

Answer 2

Can you tell me in which chapter this question belong