Find two integers b (one negative and one positive) such that 7 = b (mod 3)

Question

I am a beginner trying to really grasp this concept. How should i go along to solving this?

Answer 1

Let $a,b$ be integers such that $a \equiv b\text{ mod }n$. So, $$a-b = nk$$ for some $k \in \mathbb{Z}$. If $a$ and $n$ are given, then you need to solve

$$b = a- nk.$$

First, if you want $b > 0$, then $a - nk > 0$, or, $a > nk$.

In your case, $a = 7$ and $n = 3$. So, you need to solve $$b=7-3k$$ where $7>3k$, or, $k < \frac{7}{3} < 3$. Thus, you can take $k=1$ or $2$ (any negative $k$ works too, as pointed out). Can you continue?