Determine the remainder of $A \cdot B$ and $A+B$.

Question

$$A \equiv 3 \pmod{5}$$

$$B \equiv 4 \pmod{5}$$

• Determine the remainder of $A \cdot B$ and $A+B$.

Rewriting the equations

$$A-3 \equiv 0 \pmod{5}$$

$$B -4 \equiv 0 \pmod{5}$$

Which yields $A \in \{3,8\}$ and $B \in \{4,9\}$. As it appears, I've gone wrong somewhere due to that there are solutions more than one.

Answer 1

Just add (or multiply) and find the remainder. $$ 3 + 4 = 7 \equiv 2 \pmod{5}. $$

You get the same answer if you start from $$ 8 + 9 = 17 \equiv 2 \pmod{5}. $$

The theorem telling you that works for both multiplication and addition whatever numbers you start from is what makes modular arithmetic pretty easy (you never need to deal with numbers larger than the square of the modulus) and very useful.

Answer 2

$$A+B=5k+3+5h+4=5(k+h+1)+2\equiv 2 \text{ (mod 5) }$$

Try to do the same with $A\cdot B$.