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.

score 1 · Answer 1 · answered Sep 15 '18 at 18:00

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

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

answered Sep 15 '18 at 18:00

InsideOut

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score 1 · Accepted Answer · answered Sep 15 '18 at 18:00

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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.

answered Sep 15 '18 at 18:00

Ethan Bolker

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Could you please explain it further? – Busi Sep 15 '18 at 18:11
It's not clear what kind of further explanation you want. You can start at the wikipedia page https://en.wikipedia.org/wiki/Modular_arithmetic – Ethan Bolker Sep 15 '18 at 18:33

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

2 Answers2