CBSE Class 10 Answered
show that n^2-1 is divisible by 8, if n is an odd positive integer.
Asked by rajani.vinod | 26 May, 2019, 11:55: AM
Expert Answer
Any odd positive integer is in the form of 4p + 1 or 4p+ 3 for some integer p.
Let n = 4p+ 1,
(n2 – 1) = (4p + 1)2 – 1
Let n = 4p+ 1,
(n2 – 1) = (4p + 1)2 – 1
= 16p2 + 8p + 1
= 16p2 + 8p
= 8p (2p + 1)
⇒ (n2 – 1) is divisible by 8
(n2 – 1) = (4p + 3)2 – 1
⇒ (n2 – 1) is divisible by 8
(n2 – 1) = (4p + 3)2 – 1
= 16p2 + 24p + 9 – 1
= 16p2 + 24p + 8
= 8(2p2 + 3p + 1)
⇒ n2– 1 is divisible by 8
Therefore, n2– 1 is divisible by 8 if n is an odd positive integer.
⇒ n2– 1 is divisible by 8
Therefore, n2– 1 is divisible by 8 if n is an odd positive integer.
Answered by Arun | 26 May, 2019, 06:19: PM
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