Suppose that $n\ge2$ and $a_1,a_2,...,a_n$ are natural numbers that $ (a_1,a_2,...,a_n)=1$. Find all strictly increasing function $f: \mathbb{Z}  \to \mathbb{R} $ that:

$$ \forall  x_1,x_2,...,x_n \in  \mathbb{Z} :       f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})$$

[i]Proposed by Navid Safaei and Ali Mirzaei [/i]
1. **Initial Setup and Given Conditions:**
   We are given that \( n \ge 2 \) and \( a_1, a_2, \ldots, a_n \) are natural numbers such that \( \gcd(a_1, a_2, \ldots, a_n) = 1 \). We need to find all strictly increasing functions \( f: \mathbb{Z} \to \mathbb{R} \) that satisfy:
   \[
   \forall x_1, x_2, \ldots, x_n \in \mathbb{Z}, \quad f\left(\sum_{i=1}^{n} x_i a_i\right) = \sum_{i=1}^{n} f(x_i a_i)
   \]

2. **Simplifying the Problem:**
   Let's consider the case when \( n = 2 \). We have \( \gcd(a_1, a_2) = 1 \). We need to show that:
   \[
   f(x_1 a_1 + x_2 a_2) = f(x_1 a_1) + f(x_2 a_2)
   \]

3. **Using Bézout's Identity:**
   Since \( \gcd(a_1, a_2) = 1 \), by Bézout's identity, there exist integers \( u \) and \( v \) such that:
   \[
   a_1 u + a_2 v = 1
   \]
   This implies that for any integer \( k \), we can write:
   \[
   k = k(a_1 u + a_2 v) = k a_1 u + k a_2 v
   \]

4. **Function Properties:**
   We know that \( f \) is strictly increasing and \( f(0) = 0 \). Let's consider the function \( f \) on multiples of \( a_1 \) and \( a_2 \):
   \[
   f(k a_1) = k f(a_1) \quad \text{and} \quad f(k a_2) = k f(a_2)
   \]
   for any integer \( k \).

5. **General Case:**
   For any integer \( x \), we can write \( x \) as \( x = q a_1 + r \) where \( 0 \le r < a_1 \). Then:
   \[
   f(x) = f(q a_1 + r) = f(q a_1) + f(r) = q f(a_1) + f(r)
   \]
   Since \( f \) is strictly increasing and \( f(0) = 0 \), we can deduce that \( f \) must be linear. Let \( f(a_1) = c_1 \) and \( f(a_2) = c_2 \).

6. **Consistency Check:**
   For \( f \) to be consistent with the given functional equation, we must have:
   \[
   f(x) = c x
   \]
   where \( c \) is a constant. This is because any other form would violate the strictly increasing property or the functional equation.

7. **Conclusion:**
   Therefore, the only strictly increasing function \( f \) that satisfies the given condition is of the form:
   \[
   f(x) = c x
   \]
   where \( c \) is a positive constant.

The final answer is \( \boxed{ f(x) = c x } \) where \( c \) is a positive constant.