$\text{If}17\equiv 4\left(mod13\right)\text{then}51\equiv x\left(mod13\right)$, then find the value of $x$.

1. 12

The correct option is A 12
Given: $51\equiv x\left(mod13\right)$

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

$(a\times c)\equiv (b\times c)\left(modm\right)$

Given, $17\equiv 4\left(mod13\right)$
Hence, from the above theorem, if we multiply 3 in 17 we get 51
i.e., $17\times 3\equiv 4\times 3\left(mod13\right)$
$51\equiv 12\left(mod13\right)$
By comparing the above expression with the given one, we get $x=12$.