Modulus with divisor unknown

Question

I'm trying to find the answer to this question, but more importantly, a general formula.

$x$ is a positive integer which solves the following questions: when $280$ is divided by $x$, the remainder is $16$; when $900$ is divided by $x$, the remainder is $42$. Find the greatest possible value of $x$.

It's quite obvious that $43\leq x\leq264$, and I've converted the above into the equations $280=kx+16$ and $900=mx+42$. I'm not sure how to proceed further.

Answer 1

From $280 = kx + 16$ and $900 = mx + 42$, simple transpositions give us $858 = mx$ and $264 = kx$. Thus, $x$ is a divisor of $858$ and $264$. What is the greatest such divisor?(Think : greatest common divisor). Do you know a "general" procedure for finding the greatest common divisor?

Do you think that this satisfies the conditions of the question, and is the largest such number to do so?

Answer 2

Hint:

When $280$ is divided by $x$, the remainder is $16$. What does this tell you about the remainder when $280 - 16$ is divided by $x$?

Answer 3

The number which you are looking for has to be the HCF of $858$ and $264$ because the two conditions have to be fulfilled simultaneously. Thus the answer is $66$. You can use Euclid Algorithm or Prime factorization method to find HCF.

Note : HCF denotes Highest Common Factor