But can you Multiply?

For the last digit of the product $15 \square \color{#D61F06}4$ to be $\color{#D61F06}4$ , $\square$ can either be $2$ or $8$ . Since $\boxed 8 \boxed 8 \times 1 \boxed 8 = 15\boxed 84$ , $\implies \square = \boxed 8$ .

$(10\square + \square)\times(10+\square) = 1504 + 10\square$ $100\square + 10\square^{2} + 10\square + \square^{2} = 1504 + 10\square$ $\Rightarrow 11\square^{2} + 100\square - 1504 = 0 \text{ Now we have a quadratic so the quadratic formula seems appropriate.}$ $\Rightarrow \square = \frac{-100\pm \sqrt{100^2 -4\times 11\times -1504}}{2\times 11}$ $\text{ This gives us } \\ \square = -\frac{188}{11} \text{ or } \square = 8 \text{ but since } \square \text{ is a natural number } \\ \Rightarrow \square = \boxed{8}$ .

What digit times itself has a 4 in the units?

2 * 2 = 4 but 22 * 12 = 264

8 * 8 = 64 and 88 * 18 = 1584

$15\square 4$ is a multiple of $\square\square$ , which is a multiple of $11$ . Hence $\square=8$