(1) Show that for all odd natural numbers n both 9n + 1n and 7n + 3n are multiples of 10.

(2) Show that for all even natural numbers n both 8n - 2n and 6n - 4n are multiples of 10.

(3) Find and prove similar results to (1) for even powers.

(4) Find and prove similar results to (2) for odd powers.


NOTES AND BACKGROUND
There are eight results to be proved. You are invited to spot patterns, to make conjectures and then to prove your conjectures.

The two principal methods of proof involve ideas central to mathematics courses in the last two years of school. Finding both methods is one of the challenges here.

If you are proficient you won't need to prove all 8 cases once you have demonstrated that you can prove one, but if you do not find the proofs easy then it will be good practice to prove the results in all the cases.