Divisibility Algebra Question

[Destro]

Hi, I have been trying to get started with this question for a couple of days now. I know the rules say to show what you have attempted, or at least your progress work, but I really have no idea on how to get started.

The question is a proof,
If 2|(n^2 - 1) then, 4|(n^2 - 1) must also be true.

So I know that this is true for any odd number value for 'n', but how would I go about explaining it?
Thanks!

 

[Deleted member 4993]

Hi, I have been trying to get started with this question for a couple of days now. I know the rules say to show what you have attempted, or at least your progress work, but I really have no idea on how to get started.

The question is a proof,
If 2|(n^2 - 1) then, 4|(n^2 - 1) must also be true.

So I know that this is true for any odd number value for 'n', but how would I go about explaining it?
Thanks!

Assume that n = 2m+1 (for any m)

Now you have 'n' is an odd number

 

[Destro]

Wait, substitute n^2 with 2m+1 or n?
Also how would I justify saying that n should be an odd number?

 

[Deleted member 4993]

Wait, substitute n^2 with 2m+1 or n?
Also how would I justify saying that n should be an odd number?

I said assume:

n (not n²) = 2m+1 → n² - 1 = 4m² + 8m = 4 * (m + 2)

For any positive integer (including 0) m, you'll have odd 'n'.

m = 0 → n = 1
m = 1 → n = 3
m = 2 → n = 5 .... and so on....

 

[Ishuda]

Hi, I have been trying to get started with this question for a couple of days now. I know the rules say to show what you have attempted, or at least your progress work, but I really have no idea on how to get started.

The question is a proof,
If 2|(n^2 - 1) then, 4|(n^2 - 1) must also be true.

So I know that this is true for any odd number value for 'n', but how would I go about explaining it?
Thanks!

Factor the n²-1. Now if 2 divides n²-1, it must divide one of the factors. If it divides one of the factors then n must be (which, even or odd?) and ...