Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that  $m + a_p - a_s$ is divided by $n$.
To solve this problem, we need to find the least positive integer \( k \) such that among any \( k \) arbitrary integers \( a_1, a_2, \ldots, a_k \), there exist two numbers \( a_p \) and \( a_s \) (with \( p \neq s \)) such that \( m + a_p - a_s \) is divisible by \( n \). 

1. **Define the Greatest Common Divisor (GCD):**
   Let \( d = \gcd(m, n) \). This means that \( m \) and \( n \) can be expressed as \( m = d \cdot m' \) and \( n = d \cdot n' \) where \( \gcd(m', n') = 1 \).

2. **Consider the Circle Representation:**
   Draw \( n \) points around a circle and label them \( 0, 1, 2, \ldots, n-1 \). Connect each point \( i \) to the point \( (i + m) \mod n \). This will form \( d \) polygons, each with \( \frac{n}{d} \) vertices.

3. **Non-Divisibility Condition:**
   The condition \( a_i - a_j \) is not divisible by \( n \) implies that no two selected points can be the same modulo \( n \).

4. **Polygon Selection:**
   In each of the \( d \) polygons, we can select at most \( \left\lfloor \frac{n}{2d} \right\rfloor \) points such that no two points are consecutive. This is because if we select more than \( \left\lfloor \frac{n}{2d} \right\rfloor \) points, there will be at least two points \( a_p \) and \( a_s \) such that \( m + a_p - a_s \) is divisible by \( n \).

5. **Total Points Selection:**
   Since there are \( d \) polygons, the total number of points we can select is \( d \cdot \left\lfloor \frac{n}{2d} \right\rfloor \).

6. **Conclusion:**
   Therefore, the least positive integer \( k \) that satisfies the given condition is \( d \cdot \left\lfloor \frac{n}{2d} \right\rfloor \).

The final answer is \( \boxed{d \cdot \left\lfloor \frac{n}{2d} \right\rfloor} \).