Number theory (deductive proof)

[darklite]

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.

 

[StatusX]

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.

 

[darklite]

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]

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.