Question

$\text{If}9\equiv 3\left(mod6\right)\text{and}14\equiv 2\left(mod6\right)$, then $23\equiv x\left(mod6\right)$. Find x.

• A. x = 6
• B. x = 5
• C. x = 3
• D. x = 1

Answer 1

The correct option is B x = 5

Given: $23\equiv x\left(mod6\right)$

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

Given, $9\equiv 3\left(mod6\right)\text{and}14\equiv 2\left(mod6\right)$
Hence, from the theorem (i),
$(9+14)\equiv (3+2)\left(mod6\right)$
$\Rightarrow$ $23\equiv 5\left(mod6\right)$
By comparing the above expression with the given one, we get x = 5.