I am wondering if there is a structured way to solve this kind of problem:
There is a number $n$
- $n$ divided by $m$ (m is not given) has remainder 5
- $n$ divided by $m+1$ has remainder 1
- $n$ divided by $m+2$ has remainder 7
- $n$ divided by $m+3$ has remainder 1
- $n$ divided by $m+4$ has remainder 1
- $n$ is between 300 and 500
and (only use it if you really need them):
- $n$ divided by $m+5$ has remainder 7
- $n$ divided by $m+6$ has remainder 5
There is (at least one) solution to this puzzle, I am not sure if there are more than one. but how to find it?
From the given, can I deduce that $n$ is odd?
But is there more to know about $n$ and are there ways to find $n$ that do not rely on brute force?