Proof: Let m any odd integer.
The definition of even integer gives,
m=2p
Here, p is also integer.
According to the question,
3m + 5=3(2p) + 5
=6p + 5
=2(3p) + 2(2) + 1
=2(3p + 4) + 1
Let, (3p + 4)=a
Here, a is integer.
The above relation implies that,
3m + 5=2a + 1
The above relation 3m + 5=2a + 1 implies the definition of odd integer.
Therefore, for all integers m, if m is even then 3m + 5 is odd.