Question

# Prove that for all integers m, if even then 3m + 5 is odd.

Transformation properties
Prove that for all integers m, if even then $$3m\ +\ 5$$ is odd.

2021-01-09
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.