Find all the numbers of $5$ non-zero digits such that deleting consecutively the digit of the left, in each step, we obtain a divisor of the previous number.
To solve the problem, we need to find all 5-digit numbers such that deleting the leftmost digit at each step results in a number that is a divisor of the previous number. Let's denote the 5-digit number as $\overline{abcde}$, where $a, b, c, d, e$ are non-zero digits.

1. **Step 1: Analyze the condition for divisibility**
   - We need $\overline{bcde} \mid \overline{abcde}$.
   - This implies $\overline{bcde} \mid 10^4a + \overline{bcde}$.
   - Therefore, $\overline{bcde} \mid 10^4a$.

2. **Step 2: Determine the possible values of $a$**
   - Since $\overline{bcde}$ is a 4-digit number, it must be a divisor of $10^4a$.
   - The maximum value of $\overline{bcde}$ is 9999, so $10^4a \geq 9999$.
   - This implies $a \geq 2$ (since $10^4 \cdot 1 = 10000$ is the smallest 5-digit number).

3. **Step 3: Check each possible value of $a$**
   - For $a = 1$: $10^4 \cdot 1 = 10000$. $\overline{bcde}$ must be a divisor of 10000. However, $\overline{bcde}$ must be a 4-digit number, and no 4-digit number can divide 10000.
   - For $a = 2$: $10^4 \cdot 2 = 20000$. $\overline{bcde}$ must be a divisor of 20000. However, no 4-digit number can divide 20000.
   - For $a = 3$: $10^4 \cdot 3 = 30000$. $\overline{bcde}$ must be a divisor of 30000. However, no 4-digit number can divide 30000.
   - For $a = 4$: $10^4 \cdot 4 = 40000$. $\overline{bcde}$ must be a divisor of 40000. However, no 4-digit number can divide 40000.
   - For $a = 5$: $10^4 \cdot 5 = 50000$. $\overline{bcde}$ must be a divisor of 50000. Let's check possible values of $\overline{bcde}$:
     - $\overline{bcde} = 3125$ (since $5^4 = 3125$ and $3125 \mid 50000$).
     - Therefore, $\overline{abcde} = 53125$.
   - For $a = 6$: $10^4 \cdot 6 = 60000$. $\overline{bcde}$ must be a divisor of 60000. However, no 4-digit number can divide 60000.
   - For $a = 7$: $10^4 \cdot 7 = 70000$. $\overline{bcde}$ must be a divisor of 70000. However, no 4-digit number can divide 70000.
   - For $a = 8$: $10^4 \cdot 8 = 80000$. $\overline{bcde}$ must be a divisor of 80000. However, no 4-digit number can divide 80000.
   - For $a = 9$: $10^4 \cdot 9 = 90000$. $\overline{bcde}$ must be a divisor of 90000. Let's check possible values of $\overline{bcde}$:
     - $\overline{bcde} = 1125$ (since $9 \cdot 5^3 = 1125$ and $1125 \mid 90000$).
     - Therefore, $\overline{abcde} = 91125$.
     - $\overline{bcde} = 5625$ (since $9 \cdot 5^4 = 5625$ and $5625 \mid 90000$).
     - Therefore, $\overline{abcde} = 95625$.

4. **Step 4: Verify the solutions**
   - For $\overline{abcde} = 53125$:
     - $\overline{bcde} = 3125$, $\overline{cde} = 125$, $\overline{de} = 25$, $\overline{e} = 5$.
     - Each step results in a divisor of the previous number.
   - For $\overline{abcde} = 91125$:
     - $\overline{bcde} = 1125$, $\overline{cde} = 125$, $\overline{de} = 25$, $\overline{e} = 5$.
     - Each step results in a divisor of the previous number.
   - For $\overline{abcde} = 95625$:
     - $\overline{bcde} = 5625$, $\overline{cde} = 625$, $\overline{de} = 25$, $\overline{e} = 5$.
     - Each step results in a divisor of the previous number.

Conclusion:
The numbers that satisfy the given condition are $53125$, $91125$, and $95625$.

The final answer is $\boxed{53125, 91125, 95625}$