Question

$\text{If}17\equiv 4\left(mod13\right)\text{and}42\equiv 3\left(mod13\right)$, then _____.

• A. $59\equiv 7\left(mod13\right)$
• B. $24\equiv 7\left(mod13\right)$
• C. $59\equiv 4\left(mod13\right)$
• D. $59\equiv 5\left(mod13\right)$

Answer 1

The correct option is A $59\equiv 7\left(mod13\right)$

By the theorem on modulo operations, if $a,b,c$ and $d$ are integers and $m$ is a positive integer such that $a\equiv b\left(modm\right)$ and $c\equiv d\left(modm\right),$ then
(i) $(a+c)\equiv (b+d)\left(modm\right)$
(ii) $(a-c)\equiv (b-d)\left(modm\right)$
(iii) $(a\times c)\equiv (b\times d)\left(modm\right).$

Given that $17\equiv 4\left(mod13\right)$ and $42\equiv 3\left(mod13\right).$
Using the theorem, we have
$(17+42)\equiv (4+3)\left(mod13\right)$
$\Rightarrow$ $59\equiv 7\left(mod13\right)$