Handshaking lemma proving r-regular graph

Question

I'm trying to prove this using the handshaking lemma, but I'm not sure if this would suffice. I am a novice at writing proofs and would appreciate any type of help.

Answer 1

The number of edges, as you said is $\frac{1}{2} \sum_{v \in V} d_G(v) = \frac{7n}{2}$ where $n$ is the number of vertices. As the number of edges is an integer, $\frac{7n}{2}$ must be an integer which implies that $n$ is even which implies that $\frac{n}{2} = m$ is an integer which implies that the number of edges is $7m$ for some integer $m$.