Note that $2024 = 44 \cdot 46$. A $44 \times 46$ rectangle will have perimeter $2(44 + 46) = 180$. It is straightforward to check the other possible dimensions to show that this gives the rectangle with the least possible perimeter:
* $23 \times 88$ gives a perimeter of $2(23 + 88) = 222$.
* $22 \times 92$ gives a perimeter of $2(22 + 92) = 228$.
* $11 \times 184$ gives a perimeter of $2(11 + 184) = 390$.
* If one of the dimensions is $1$, $2$, $4$, or $8$, then the other dimension is greater than $200$, yielding rectangles with greater perimeters.
Thus the least possible perimeter is $180$.