I KNOW THIS IS SOLUTION BUT I DON'T KNOW WHY?
We first find the difference of the numbers and then find the HCF of the got numbers.
183−91=92
183−43=140
91−43=48
Now find HCF of 92, 140 and 48, we get
92=2×2×23
140=2×2×5×7
48=2×2×2×2×3
HCF(92, 140, 48) = 4
Therefore, 4 is the required number.
Can you explain how we get correct answer using this method.
Call the greatest number $n$ and the common remainder $x$, so our problem is
$$\begin{align} x&\equiv 43\pmod{n}\\ x&\equiv 91\pmod{n}\\ x&\equiv 183\!\!\!\pmod{n} \end{align}\qquad$$
By general CRT theory this system is solvable iff pairwise solvable, i.e. iff
$$\begin{align} &n\mid 91\!-\!43,\, 183\!-\!91,\, 184\!-\!43\\ \iff \ \ &n\mid 48,92,140\\ \iff\ \ &n\mid \gcd(48,92,140) = 4\end{align}\ \ $$
where the final arrow is by the gcd Universal Property.
