Question

# Find the value of $x$ that satisfying $150x \equiv 35(Mod31)$ is$\ A)14 \\ B)22 \\ C)24 \\ D)12 \\ \$

Hint: To proceed with this solution we have to compare the given expression with the standard form.

Given expression is $150x \equiv 35(Mod31)$
Let us compare the expression with standard form i.e. $a \equiv b(\bmod n)$
Before comparing it lets us know what does the standard form means
Here,
$a \equiv b(\bmod n)$ Means $a - b$ is divisible by $n$
So, here if we compare we can say $a = 150x,b = 35$ and $n = 31$
So,
$150x \equiv 35(Mod31)$
$\Rightarrow 150x - 35$ is divisible by $31$
$\Rightarrow 5(30x - 7)$ is divisible by $31$
Now here let us say that $30x - 7 = 31k$ where $k$is any natural number.
From this we can equate x value as
$\Rightarrow$ $x = \dfrac{{31k + 7}}{{30}}$
$\Rightarrow \dfrac{{30k + (k + 7)}}{{30}}$
$\Rightarrow \dfrac{{30k}}{{30}} + \dfrac{{k + 7}}{{30}}$
$\Rightarrow k + \dfrac{{k + 7}}{{30}}$
Here $k + 7$ is divisible by $30$ where $k = 23$ which is smallest possible value
So, if $k = 23$ then
$\Rightarrow k + \dfrac{{k + 7}}{{30}}$=$k + \dfrac{{23 + 7}}{{30}} = k + \dfrac{{30}}{{30}} = k + 1$
And we know that $k = 23$
So, here
$\ \Rightarrow x = K + 1 \\ \Rightarrow x = 23 + 1 \\ \Rightarrow x = 24 \\ \$
From this we can say that for $x = 24$ is the value that satisfying the expression
Option C is the correct

NOTE: Problems like the above model have to be compared with standard forms and in the above problem we have to substitute k value in two places in the same equation. Instead of substituting K value in the two places at a time it is better to substitute in one place then make the equation simple so get the answer directly.