Question

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

• A. $-25\equiv -1\left(mod13\right)$
• B. $-35\equiv 1\left(mod13\right)$
• C. $-25\equiv 1\left(mod13\right)$
• D. $59\equiv 6\left(mod13\right)$

Answer 1

The correct option is C $-25\equiv 1\left(mod13\right)$

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) $(a-c)\equiv (b-d)\left(modm\right)$ and
(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 (ii), we have
$(17-42)\equiv (4-3)\left(mod13\right).$
$\Rightarrow$ $-25\equiv 1\left(mod13\right)$