Prove that if a is an odd integer, then 24 | a(a^2-1).
Proof. from a previous in my homework, I know that the square of an odd integer can be written in the form of 8k + 1.
Now, let a = 2w + 1 and a^2 = 8k + 1, for some integers w and k.
Then
But now, how do I factor out 24?
I prefer to do it this way. Notice that and are consecutive even integers since a is odd. Hence the product is divisible by 8. (NB: the product of two consecutive even intgers and is . One of and is even and so must be divisible by 8.)
Also are three consecutive integers and so one of them must be a multiple of 3.
Putting the two together, we have that the product is divisible by both 8 and 3. Now the proof is almost finished. I’ll let you finish it yourself.