Question

m
${m - }^{2} is \: divisible \: by \: 8 \: if \: n \: is \\ \\ (a)an \: integer \\ (b)a \: ntural \: number \\ (c)na \: odd \: integer \\ (d) \: an \: even \: intger$
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Answer 1

Step-by-step explanation:

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Class 6

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>>Divisibility Rules

>>n^2 - 1 is divisible by 8 , if n is numb

Question

n

2

−1 is divisible by 8, if n is ___ number.

Medium

Solution

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Any odd positive integer is in the form of 4p + 1 or 4p+ 3 for some integer p.

Let n=4p+1,

(n

2

–1)=(4p+1)

2

–1=16p

2

+8p+1=16p

2

+8p=8p(2p+1)

⇒(n

2

–1) is divisible by 8.

(n

2

–1)=(4p+3)

2

–1=16p

2

+24p+9–1=16p

2

+24p+8=8(2p

2

+3p+1)

⇒n

2

–1 is divisible by 8.

Therefore, n

2

–1 is divisible by 8 if n is an odd positive integer.