Question: Ali have 100 knife and 2020 people to murder. He name the 2020 people as 1,2,3,\ldots,2020 in a circle. First time, he stabbed all the person 1,2,3,\ldots,2020 exactly 1 time. Then 2nd time, he stabbed the person 2,4,6,8,\ldots,2020 exactly another 1 time. At $i$ th time, Ali stabbed person have numbers of multiple $i$ with exactly one time each in each operation. By continue this process, how many total of time Ali stabbed?
1. **State the problem:**
Ali stabs $2020$ people arranged in a circle. On the $i$-th round, he stabs every person whose number is a multiple of $i$ exactly once. We want to find the total number of stabs after all rounds from $i=1$ to $i=2020$.
2. **Understanding the problem:**
- On round 1, he stabs all $2020$ people (all multiples of 1).
- On round 2, he stabs all even-numbered people (multiples of 2).
- On round $i$, he stabs all multiples of $i$.
3. **Formula and approach:**
The total number of stabs is the sum over $i=1$ to $2020$ of the count of multiples of $i$ in $1$ to $2020$.
The number of multiples of $i$ from $1$ to $2020$ is:
$$\left\lfloor \frac{2020}{i} \right\rfloor$$
So total stabs:
$$S = \sum_{i=1}^{2020} \left\lfloor \frac{2020}{i} \right\rfloor$$
4. **Calculate the sum:**
We need to compute:
$$S = \sum_{i=1}^{2020} \left\lfloor \frac{2020}{i} \right\rfloor$$
This sum counts how many times each person is stabbed because each person $k$ is stabbed once for every divisor $i$ of $k$.
5. **Interpretation:**
This sum is also the total number of divisors of all numbers from $1$ to $2020$ combined, because each divisor corresponds to one stab.
6. **Final answer:**
The total number of stabs Ali made is:
$$\boxed{\sum_{i=1}^{2020} \left\lfloor \frac{2020}{i} \right\rfloor}$$
This is the exact formula for the total stabs.
If you want a numerical value, you can compute this sum by evaluating each term and adding them up.