Basic algebra. Please help. When $1200$ is divided by $N$ and $N+x$, the remainder in both cases is $35$.

What is the maximum value of x?
What can we say about $1200-35=1165$ in relation to $N$ and $N+x$? Particularly with regards to divisibility.

Hint 2: Look at all factors of $1165$, see what you can say.
I will use Haskell to brute-force this problem. Here's an extremely brief GHCi session:

Prelude&gt; [ n | n &lt;- [1..1200], 1200 `mod` n == 35]
[233,1165]


Because $1200 = (233 \times 5) + 35 = 1165 + 35$. Since $1165-233 = 932$, we have $x = 932$. Alternatively, we could have used a sexier list comprehension, as follows:

Prelude&gt; [ x | n &lt;- [1..1200], x &lt;- [1..(1200-n)], 1200 `mod` n == 35, 1200 `mod` (n+x) == 35]
[932]


The interpretation of these "experimental" results is left to the reader.