# Number theory (deductive proof)

I just started learning gr. 12 discrete math a few days and I’m already having trouble with two very similar questions…
Using deductive proof
1) Prove that if 4 is subtracted from the square of an integer greater than 3, the result is a composite number.
2) Prove that if 25 is subtracted from the square of an odd integer greater than 5, the resulting number is always divisible by 8.

I started 1) by x2-4 = composite number, x > 3 I realized I could factor it down to (x-2)(x+2) = composite number, but I got lost from there.

Then I started 2) in a similar manner by (x2-25)/8. However I’m not sure if the equation is correct so I stopped there.

As you can tell, I’m not exactly the best at deductive proving. So thanks in advance. Related Calculus and Beyond Homework Help News on Phys.org
StatusX
Homework Helper
For 1), it looks like you're done. What do you think you're missing?

For 2), write the odd integer as 2k+1, where k is now any integer greater than 2. Plug in and simplify.

1) thnx...i just remembered that a composite number is a number that could be factored

2) so i factored (x^2 - 25)/ 8 to (x-5)(x+5)/28
and i plugged in 2k + 1 into the equation so (2k+1-5)(2k+1+5)/8
(2k+4)(2k+6)/8
2(k+2)2(k+3)/8
then i'm not sure about how to prove that is divisible by 8

also i just found another question I'm not so sure about

3) Prove that n^5-5n^3+4n is divisible by 120 for all positive integers n is greater than or equal to 3.

At first I factored it to n(n^4 - 5n^2 +4)
n(n^2-4)(n^2-1)
Then I wasn't sure about how to prove it from there...

StatusX
Homework Helper
First, there's no need to write that "/8". Second, it seems that all you have left is to show (k+2)(k+3) is even. Can you do this? And for 3), try factoring a little more using the difference of squares formula.