Thread: Discrete Math Help

I am new to Discrete Math. I am able to figure out how to do 1, 2, and 4 of the attachment, but do not have a clue where to start on the rest of the problems. Any help at all would be great. Thanks in advance.

Hello, Hornedog20!

3. What is the largest divisor of $\displaystyle n^7 - n^3$ for all $\displaystyle n \in N$ ?
Are you sure of the wording?
The largest divisor of $\displaystyle {\color{blue}n^7-n^3}$ is, of course: .$\displaystyle {\color{blue}n^7-n^3}$

Factor: .$\displaystyle n^7-n^3 \;=\;n^3(n^4-1) \;=\;n^3(n^2-1)(n^2+1) \;=\;n^3(n-1)(n+1)(n^2+1) $

Then $\displaystyle n^2+1$ is the largest prime factor.

Thanks for the reply. That is what I was thinking for #3 as well. I believe I have figured out 5, 6, and 7 now as well. I still am having problems with 8 and 9 though.