Why do this problem?
As well as offering practice in working with indices, the proofs in this problem can be done using several different techniques, using the binomial theorem or modular arithmetic.
See Power Mad!
for a related problem that you may wish to set students instead of or as well as this one.
Start by inviting students to explore $9^n+1^n$ for odd values of $n$. When they observe that the answers all seem to be multiples of $10$, give students a short while to come up with their own convincing arguments. They are likely to use an argument based on the last digits of odd powers of $9$, and this could be a good opportunity to introduce the language and notation of modular arithmetic,
and the idea of working mod $10$. This article
could be given to students as background reading.
Alternatively, if you wish to focus on using the binomial theorem, invite students to express the questions in terms that can be expanded: for example, rewriting $7^n + 3^n$ as $7^n + (10-7)^n$. Students can write out the expression for odd and even values of $n$ and construct a convincing justification why the expression is always a multiple of $10$ when $n$ is odd.
What happens to the units digit when you find consecutive powers of a number?
Can we use the fact that the numbers occurring in each part add up to 10?
For some challenging work on powers, see Giants
provides a slightly more playful introduction to the ideas met in this problem.