Problem Definition:
The integer \(X\) must satisfy the condition that for any integer \(Y\), the remainder of \(Y\) divided by \(X\) is always 1. This can be expressed as:
\[ Y \mod X = 1 \]
Properties of \(X\):
Given that \(X\) must be within the range of 2 to 40, it follows that \(X\) is not divisible by any integer from 2 to 40. Mathematically, \(X\) must be co-prime with all integers from 2 to 40.
Potential Values for \(X\):
The largest integer meeting this criterion is a prime number less than 40, which is not divisible by any number between 2 and 40. The largest prime number below 40 is 37.
Confirmation:
If \(X = 37\), then any integer \(Y\) divided by 37 will yield a remainder of 1:
\[ Y = k \cdot 37 + 1, \quad k \in \mathbb{Z}. \]
Therefore, \(X\) is determined to be 37.