# What's the answer of (10+13) ≡?

As to Modulus operation I only have seen this form:

(x + y) mod z  ≡ K


So I can't understand the question, by the way the answers are :

a) 8 (mod 12)
b) 9 (mod 12)
c) 10 (mod 12)
d) 11 (mod 12)
e) None of the above

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mod is more than just an operation, it signifies an equivalence relation (the relation of having the same remainder after division by the modulus). Which of the given answers has the same remainder as 10+13=23 when divided by 12? [Also, please read tag descriptions so that you don't use incorrect ones; modules are not relevant to this question though it sounds related.] – anon Sep 4 '12 at 3:25
10+13 = 23. so, what is $23 \pmod {12}$? well the remainder when dividing 23 by 12 is 11 (23 = 12 + 11) so the answer is d – Deven Ware Sep 4 '12 at 3:25

The OP may be taking a Computer Science course, in which if $b$ is a positive integer, then $a\bmod{b}$ is the remainder when $a$ is divided by $b$. In that case $\bmod$ is a binary operator. That is different from the $x\equiv y\pmod{m}$ of number theory, which is a ternary relation (or, for fixed $m$, a binary relation).

Calculating $(10+13)\bmod{12}$ is straightforward. Find $10+13$, and calculate the remainder on division by $12$. We get $(10+13)\bmod{12}=11$. So $(10+13)\bmod{12} \equiv 11\pmod{12}$.

Remark: It seems unusual to use the binary operator and the ternary relation in a single short expression.

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It is d).

$23 \equiv 11 \text{ mod } 12$ since $(12)(1) + 11 = 23$. In other word, the remainder upon division by $12$ is $11$.

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