$\begingroup$

I am enrolled in Discrete Mathematics 2 and I am having trouble understand a lot of the material. For the particular problems I need help with I need to:

Prove each of the given statements, assuming that $a,b,c,d$, and $n$ are integers with $n>1$ and that $a\equiv c\bmod n$ and $b\equiv d\bmod n$.

These are the statements:

1. a. $a+b\equiv c+d\bmod n$
   b. $a-b\equiv c-d\bmod n$
2. $a^2\equiv c^2\bmod n$
3. $a^m\equiv c^m\bmod n$ for all integers $m\ge1$ (Use mathematical induction on $m$).

I am in pretty bad shape as far as approaching and completing these problems and I need this to be explained to me as simply as possible (as if I am a 3 year old). Thank you all for your help in advanced.

$\endgroup$ 3

1 Answer

$\begingroup$

I'll do the first one for you. The same logic follows for the rest. $a\equiv c \pmod n$ means $a=c+mn$, where $m\in\mathbb{Z}$ and similarly for $b=d$ mod$n$. So $(c+d)=(a+m_1n)+(b+m_2n)=(a+b)+(m_1+m_2)n\equiv(a+b) \pmod n$

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy