Question

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

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Answer 1

Answer:answer is 3

Step-by-step explanation:n^2-1 =8

n^2 =8+1=9

so send square on other side we get

n = under root 9

so root of 9 is 3

n = 3

Answer 2

Solution :

Let n be any positive odd  integer

n = bq + r

b = 4

Substituting the value of b in a = bq + r

Odd Positive Integers are = 1 , 3 , 5 , 7 , 9 ........

Putting the value in place of remainder

n= 4q + 1

n = 4q + 3

Case  I

If n = 4q + 1

When ,

n^2 - 1

⇒ (4q + 1)^2  - 1

⇒ 16q^2 + 8q + 1 - 1

⇒ 8q ( 2q + 1 )

8q (2q + 1 ) is divisible by 8

Case II

If n = 4q + 3

When ,

n^2 - 1

⇒ (4q + 3) - 1

⇒ 16q^2 + 24q + 9 - 1

⇒ 8(2q^2 + 3q + 1 )

8(2q^2 + 3q + 1 ) is divisible by 8

Therefore , for an odd positive integer the value of n^2 -1 is divisible by 8