Question

show that n3-n is divisible by 8 if n is an odd number

Answer 1

Given:

A number n and n³ - n. where n is odd.

To Show :

That n³-n is divisible by 8 .

Solution:

Its given that n is an odd  number.

Any odd number can be represented in the form 2p +1, p = 0,1,2,...

Therefore,

• n = 2p+1
• n³ = (2p+1)³ = (2p)³ + 3(2p)² + 3(2p) + 1
• n³ = 8p³ + 12p² + 6p + 1

Therefore,

• n³ - n = 8p³ + 12p² + 6p + 1 -2p -1
• n³ - n = 8p³ + 12p² + 4p
• n³ - n = 8p³ + 4p(3p + 1)

Let p be an odd number,

• 3 x odd number  = odd number
• 3 x odd number + 1 = even number
• Therefore 3p + 1 is even when p is odd

Therefore,

• p(3p+1) is even when p is odd
• p(3p+1) is even when p is even,

Therefore 4p(3p+1) is divisible by 4 and p(3p+1) is divisible by 2.

Therefore 4p(3p+1) is divisible by 8.

• n³ - n = 8( p³ + p(3p+1)/2)

Thus proved that n³-n is divisible by 8.