We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$. We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$, thus defining the sets $M_1,M_2,...,M_n$. Find the minimum value of $n$.
1. **Understanding the Problem:**
   We have 160 sets \( A_1, A_2, \ldots, A_{160} \) where the size of each set \( A_i \) is \( i \). We need to create new sets \( M_1, M_2, \ldots, M_n \) by removing the same number of elements from some of the sets \( A_i \) in each step. The goal is to determine the minimum number of steps \( n \) required to empty all the sets \( A_1, A_2, \ldots, A_{160} \).

2. **Binary Representation Approach:**
   We can use the binary representation of the indices \( i \) to determine the steps. Each number \( i \) from 1 to 160 can be represented in binary form, which requires at most 8 bits (since \( 2^7 = 128 \) and \( 2^8 = 256 \)).

3. **Step-by-Step Removal:**
   In each step \( k \) (where \( k = 0, 1, 2, \ldots, 7 \)), we remove \( 2^k \) elements from each set \( A_i \) whose binary representation has the \( k \)-th bit equal to 1. This ensures that in each step, we are removing elements in a way that corresponds to the binary representation of the indices.

4. **Ensuring All Sets are Emptied:**
   By following this procedure, we ensure that all sets \( A_i \) will be emptied after 8 steps. This is because each bit position from 0 to 7 will be addressed exactly once, and the sum of the elements removed in each step will equal the total number of elements in each set.

5. **Contradiction for Fewer Steps:**
   Suppose we could empty all sets in fewer than 8 steps, say in 7 steps. In 7 steps, we can create at most \( 2^7 = 128 \) different patterns of participation and non-participation. However, we have 160 sets, which means that at least two sets must participate in exactly the same steps. Since these two sets start with different numbers of elements, one of them will not be empty at the end, leading to a contradiction.

6. **Conclusion:**
   Therefore, the minimum number of steps \( n \) required to empty all the sets \( A_1, A_2, \ldots, A_{160} \) is 8.

The final answer is \( \boxed{8} \).