Trying to compute the greatest value of $x$.

Question

$a,b,c,x,k$ are positive natural numbers.

$$86 = ax+k$$

$$142 = bx + k$$

$$252 = cx+k$$

I'm trying to compute the greatest value of $x$.

Let's assume $ k = 1$ (we want x to take its greatest value), we have that

$$85 = ax$$

$$141 = bx$$

$$251 = cx$$ However, this makes no literal sense. Could you assist me?

Regards

Answer 1

First note that $x$ cannot be any larger than gcd$(142-86, 252-142)$ $=$ gcd$(56,110)$ $=2$.

Yet 2 works: $a=42,b=70,c=125, k=2$.