Prove that if 7n+4 is even and n is a positive integer, then n is even.

Solution:

Given 7n + 4 is even

Let 7n + 4 = 2m, for some integer ‘m’.

Then, 7n + 4 = 2m

7n = 2m - 4

n = 2 × (m - 2)/7, which is multiple of 2

Therefore n is even.

Note: Conversely, if n is even then n = 2m for some integer m

Now, 7n + 4 = 7(2m) + 4

= 2(7m + 2)

= 2(some integer)

∴ 7n + 4 is even integer

Prove that if 7n+4 is even and n is a positive integer, then n is even.

Summary:

If 7n+4 is even and n is a positive integer, then n is even.