Question -
Answer -
Solution:
(C) an odd integer
Explanation:
Let x = n2 – 1
In the above equation, n can be either even orodd.
Let us assume that n= even.
So, when n = even i.e., n = 2k, where k is aninteger,
We get,
⇒ x = (2k)2-1
⇒ x = 4k2 – 1
At k = -1, x = 4(-1)2 – 1 = 4– 1 = 3, is not divisible by 8.
At k = 0, x = 4(0)2 – 1 = 0 –1 = -1, is not divisible by 8
Let us assume that n= odd:
So, when n = odd i.e., n = 2k + 1, where k isan integer,
We get,
⇒ x = 2k + 1
⇒ x = (2k+1)2 – 1
⇒ x = 4k2 + 4k + 1 – 1
⇒ x = 4k2 + 4k
⇒ x = 4k(k+1)
At k = -1, x = 4(-1)(-1+1) = 0 which isdivisible by 8.
At k = 0, x = 4(0)(0+1) = 0 which is divisibleby 8 .
At k = 1, x = 4(1)(1+1) = 8 which is divisibleby 8.
From the above two observation, we canconclude that, if n is odd, n2-1 is divisible by 8.
Hence, option (C) is thecorrect answer.