Prove that if n is positive integer, then n is odd if and only if 5n + 6 is odd.

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First Part:

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.

Second Part:

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.

Final Decision:

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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