Show that x4 – 1 is divisible by 16 whenever x is an odd integer.
Prove x4 – 1 is divisible by 16 if x is odd. Let x = 2k + 1, k ≥ 0.
x4 – 1 = (2k + 1)4 – 1
= (2k + 1)2(2k + 1)2 – 1
= (4k2 + 4k + 1)(4k2 + 4k + 1) – 1
= 16k4 + 16k3 + 4k2 + 16k3 + 16k2 + 4k + 4k2 + 4k + 1 – 1
= 16k4 + 32k3 + 24k2 + 8k
= 16(k4 + 2k3) + 24k2 + 8k
= (4k2 + 4k + 1)(4k2 + 4k + 1) – 1
= 16k4 + 16k3 + 4k2 + 16k3 + 16k2 + 4k + 4k2 + 4k + 1 – 1
= 16k4 + 32k3 + 24k2 + 8k
= 16(k4 + 2k3) + 24k2 + 8k
All we need to worry about is 24k2 + 8k = 8k(3k + 1).
If k is even we have 8 · even · odd = 8 · 2 · something: OK.
If k is odd then 3k + 1 is even so we have 8 · odd · even = 8 · 2 · something; OK again.
OR
x4 – 1 = (x2 + 1)(x2 – 1) = (x2 + 1)(x + 1)(x – 1)
Then substitute x = 2k + 1 and proceed the same way.