Explanation:
Exp: C
an odd integer
Let a = m\(^{2}\) - 1
Here m can be ever or odd.
Case I: m = Even i.e., m = 2k, where k is an integer,
\(\Rightarrow\)a = (2k)\(^{2}\) - 1
\(\Rightarrow\)a = 4k\(^{2}\) - 1
At k = -1, = 4 (-1)\(^{2}\) - 1 = 4 - 1 = 3, which is not divisible by 8.
At k = 0, a = 4 (0)\(^{2}\) - 1 = 0 - 1 = -1, which is not divisible by 8, which is not.
Case II: m = Odd i.e., m = 2k + 1, where k is an odd integer.
\(\Rightarrow\)a = 2k + 1
\(\Rightarrow\)a = (2k + 1)\(^{2}\) - 1
\(\Rightarrow\)a = 4k\(^{2}\) + 4k + 1 - 1
\(\Rightarrow\)a = 4k\(^{2}\) + 4k
\(\Rightarrow\)a = 4k(k + 1)
At k = -1, a = 4(-1)(-1 + 1) = 0 which is divisible by 8.
At k = 0, a = 4(0)(0 + 1) = 4 which is divisible by 8.
At k = 1, a = 4(1)(1 + 1) = 8 which is divisible by 8.
Hence, we can conclude from the above two cases, if m is odd, then m\(^{2}\) - 1 is divisible by 8.
