Prove that n 2 − 1 is divisible by 8 for any odd integers n.Below...
Prove that is divisible by 8 for any odd integers n.
Below is my proof and I am confused about a few points. I am not sure the final lines are correct as I know that showing 2,4 are factors of is enough to prove that it is divisible by 8 and I have looked at some other examples. In an example of 30, both 3 and 6 are factors of 30 but 30 is not divisible by 18. But I am stuck on how to modify this proof to be complete.
To prove this statement, I intend to use direct proof. Since n has to be an odd integer as prescribed in the statement, by the definition of odd numbers, must be divisible by 8 where for some integer k. Next, we expand to be , which simplifies to .
First, we use the distributive property to get the following . We let and therefore . Hence, we know that must be divisible by 4.
Then, we use the distributive property to factor 4k from the expression to get . By the definition of even and odd number, if k is odd then must be even and if k is even then is odd. As an integer is Since 2 and 4 are common factors of , which means 8 must also be a factor of .
is therefore divisible by 8 where for some integer k. Therefore, we have proven that is divisible by 8 for any odd integers n.