Find the greatest value of $x$ satisfying $21\equiv 385\pmod x$ and $587\equiv 167\pmod x $ 
Problem: Find the greatest value of $x$ satisfying $21\equiv 385\pmod x$ and $587\equiv 167\pmod x $.
Solution: I think $21\equiv 385\pmod x$ is not possible.

There is some printing mistake in the question
Am I right?
 A: You have $x$ divides $385 - 21 = 364$ and $x$ divides $587 - 167 = 420$.
All you have to do is to find the largest number that divides both $364$ and $420$. Maybe think about the greatest common divisor?
A: Your looking for the greatest integer $x$ such that $$21\equiv 385\pmod x\quad \text{and}\quad 587\equiv 167\pmod x $$
By definition $a\equiv b \pmod x$ means that $x$ divides $a - b$, which we denote as $x\mid (a - b)$.
So we have that $$x\mid (385 - 21) = 364\quad\text{and}\quad x\mid (587 - 167) = 420$$
The greatest integer $x$ dividing $a, b$ is the greatest common divisor of $a, b$: $\gcd(a, b)$. So in this case, we're looking for $x = \gcd(364, 420)$, where $\,364 = \color{blue}{\bf 4\cdot 7} \cdot 13$ and $420 = 3\cdot \color{blue}{\bf 4}\cdot 5 \cdot \color{blue}{\bf 7}$.
Hence, $x = \gcd(364, 420) = \color{blue}{\bf 4\cdot 7} = 28$.
A: As has been mentioned
$$
21\equiv385\pmod{x}\iff x\mid364
$$
and
$$
587\equiv167\pmod{x}\iff x\mid420
$$
Since, $420-364=56$, we know that $x\mid56$.
Since $364-6\cdot56=28$, we know that $x\mid28$.
Now, if we check, $28\mid364$ and $28\mid420$.
Therefore, since $x\mid28$ and both $28\mid364$ and $28\mid420$, we see that $x=28$ is the largest number that satisfies the given equivalences.
