Lishi pressed a calculator to multiply a 3-digit number by a 1-digit number. For the 1-digit number, she made a mistake by pressing a wrong number. She obtained an incorrect product of 624 which is $\frac{3}{4}$ of the correct product. What is the smallest possible value of the 3-digit number? Ans: ___

1. Find the correct product

Since the incorrect product $624$ is $\frac{3}{4}$ of the correct product, we have:
$3 \text{ units} = 624$
$1 \text{ unit} = 624 \div 3 = 208$
$\text{Correct product} = 4 \text{ units} = 208 \times 4 = 832$

2. Determine the possible 3-digit numbers

Let the 3-digit number be $x$ and the correct 1-digit number be $y$. We know $x \times y = 832$.
We also know $x \times z = 624$, where $z$ is the wrong 1-digit number.
Thus, $x$ must be a common factor of $832$ and $624$.
$832 - 624 = 208$
So $x$ must be a factor of $208$.
The factors of $208$ are $1, 2, 4, 8, 13, 16, 26, 52, 104, 208$.
Since $x$ is a 3-digit number, the possible values for $x$ are $104$ and $208$.

3. Find the smallest possible value of the 3-digit number

If $x = 104$, then $y = 832 \div 104 = 8$ and $z = 624 \div 104 = 6$. Both $y$ and $z$ are 1-digit numbers, which is valid.
If $x = 208$, then $y = 832 \div 208 = 4$ and $z = 624 \div 208 = 3$. Both $y$ and $z$ are 1-digit numbers, which is valid.
The smallest possible 3-digit number is $104$.