Janez wants to make an $m\times n$ grid (consisting of unit squares) using equal elements of the form $\llcorner$, where each leg of an element has the unit length. No two elements can overlap. For which values of $m$ and $n$ can Janez do the task?
1. **Understanding the Problem:**
   Janez wants to tile an \(m \times n\) grid using L-shaped trominoes (each tromino covers 3 unit squares). The goal is to determine for which values of \(m\) and \(n\) this is possible.

2. **Initial Observations:**
   - Each tromino covers 3 unit squares.
   - Therefore, the total number of unit squares in the \(m \times n\) grid must be divisible by 3. This gives us the condition:
     \[
     3 \mid mn
     \]

3. **Coloring Argument:**
   - Consider a checkerboard coloring of the \(m \times n\) grid. Each tromino covers 3 squares, and it will always cover either 2 black and 1 white square or 2 white and 1 black square.
   - For the grid to be completely tiled by trominoes, the number of black squares must equal the number of white squares. This implies that the total number of squares must be even:
     \[
     2 \mid mn
     \]

4. **Combining Conditions:**
   - From the above two conditions, we have:
     \[
     3 \mid mn \quad \text{and} \quad 2 \mid mn
     \]
   - Combining these, we get:
     \[
     6 \mid mn
     \]

5. **Examples and Verification:**
   - We know that a \(2 \times 3\) grid can be tiled with trominoes:
     \[
     \begin{array}{|c|c|c|}
     \hline
     \llcorner & \llcorner & \llcorner \\
     \hline
     \llcorner & \llcorner & \llcorner \\
     \hline
     \end{array}
     \]
   - Similarly, a \(5 \times 6\) grid can be tiled with trominoes by combining smaller tilings.

6. **Generalization:**
   - By tiling smaller grids and combining them, we can tile any \(m \times n\) grid where \(6 \mid mn\).

Conclusion:
Janez can tile an \(m \times n\) grid using L-shaped trominoes if and only if \(6 \mid mn\).

The final answer is \( \boxed{ 6 \mid mn } \).