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First we will proof in (→) direction
“If n is odd then 5n + 6 is odd”
Let n be a positive odd number
Let, n = 2m + 1, where m ∈ Z+
5n + 6 = 5(2m + 1) + 6
= 10m + 5 + 6
= 10m + 11
10m is even and 11 is odd, therefore 10m + 11 is odd.
Thus, if n is odd then 5n + 6 is also odd.
Next, we will proof in (←) direction
“If 5n+6 is odd, then n is also odd”
Let, 5n + 6 = 2m + 1, where m ∈ Z+
Now, assume n = 2k (n is even) in the above statement
5n + 6 = 10k + 6
10k is even and 6 is even. Therefore, 5n + 6 is also even. But it is contradictory.
Therefore, our assumtion that n is incorrect. thus, n must be odd.
From the above 2 cases we can conclude that if n is a positive integer then n is odd if and only if 5n+6 is odd.
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