Question

Give a direct proof of the statement: "If an integer n is odd, then 5n-2 is odd"?

Answer 1

Please, see below.

Explanation:

If `n` is an odd integer, it can be written as

`n = 2k-1`

where `k` is another integer, be it even or odd. Hence,

`5n-2 = 5(2k-1) - 2 = 10k-5-2=10k-7`

10k can be written as 2`*`5k and since it has 2 as a factor, it is even. Since the difference of an even number (10k) and an odd one (7) is also odd, the statement is proven.