To Prove
n2 – n is divisible by 2 ∀ n ∈ Z
Proof
n2 – n = n * (n – 1)
For the value of n, we have 2 cases
- n is even
If n is even, then n * (n – 1) is even since multiplying even number and odd number gives even number. Therefore, n2 – n is divisible by 2 if n is even. - n is odd
If n is odd, then (n – 1) is even. This implies that n * (n – 1) is also even since multiplying even number and odd number gives even number. Therefore, n2 – n is divisible by 2 if n is odd.
Therefore, we can say that n2 – n is divisible by 2 ∀ n ∈ Z
Alternative Proof
Let n be an even number. If n is even, then we can write n as
n = 2k, ∀ k ∈ Z
n2 – n = (2k)2 – (2k)
= 4k2 – 2k
= 2 * (2k2 – k)
Therefore, n2 – n is divisible by 2 if n is even.
Let n be an odd number. If n is odd, then we can write n as
n = 2k + 1, ∀ k ∈ Z
n2 – n = (2k + 1)2 – (2k + 1)
= 4k2 + 1 + 4k – 2k – 1
= 4k2 + 2k
= 2 * (2k2 + k)
Therefore, n2 – n is divisible by 2 if n is odd.
Using the above 2 proofs, we can say that n2 – n is divisible by 2 ∀ n ∈ Z.