Every even number can be represented as 2j, where j is any integer
Every odd number can be represented as 2k+1, where k is any integer.
so, according to question:
2k + 2j+1 = 4k +1 = 2(k+j) +1 = 2m +1, where m =k+j
So, whenever we add one even number with an odd number we will always get an odd number, no matter what. The teacher has given you a trick question, and probably wanted to test your knowledge at that time.
Though this answer may be of no use to you at this time, but it may help others, so I am leaving this answer.