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# Show that: n^{2 }- 1 is divisible by 8 if n is an odd positive integer.

Odd positive integers are odd numbers greater than zero. Integers are denoted by 'z'.

## Answer: n^{2 }- 1 is divisible by 8 for n is an odd positive integer.

Let's see how we can prove this.

**Explanation: **

Let n = 4z +1 where z is an positive integer.

On substituting the value of z in n^{2 }- 1, we get

⇒ ( 4 z + 1)^{2 }- 1 = 0

⇒ 16z^{2 }+ 1 + 8 z - 1 = 0

⇒ 16z^{2 } + 8 z = 0

⇒ 8z ( 2 z + 1) = 0

Since (2z + 1) is an odd number and a multiple of 8, it is also divisible by 8.

### Thus, n^{2 }- 1 is divisible by 8 if n is an odd positive integer.

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