Question

Show that n²-1os divisible by 8,if n is an odd positive integer

Answer 1

Step-by-step explanation:

Solutions ⇒ 

To Prove ⇒ (n² - n) is divisible by 8.

Given ⇒ n is an odd positive integer.

Proof ⇒

We know that an Odd Positive Integer n is always written in the form of (4k + 1) or (4q + 3) or (4q + 5) and so on.

If n = 4k + 1 

Then, n² - 1 = (4k + 1)² - 1 

 = 16 k² + 1 + 8k - 1 

[∵ (a + b)² =  a²  + b² + 2ab]

 = 16k² + 8k

 = 8k(2k + 1)

Hence, it is divisible by 8.

If n = 4k + 3

Then, n² - 1 = (4k + 3)² - 1

 = 16k² + 9 + 24k - 1

 = 16k² + 24k + 8

 = 8(2k² + 3k + 1)

Hence, it is also divisible by 8.

If n = 4k + 5

Then, n² - 1 = (4k + 5) - 1

 = 16k² + 25 + 40k - 1

 = 16k² + 40k - 24 

 = 8(2k² + 5k - 3)

Hence, it is also divisible by 8.

Now, For any value of n, n² - 1 is always be divisible by 8.

Hence Proved.

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