Starting as normal for multiplication, 4$\times$4 = 16, so P must be 6. Both Ps are replaced by 6s in the calculation below, and the 1 that is carried (from the 16) is also shown.

Now, 6$\times$4 + 1 = 25, so N must be 5. Both Ns are replaced by 5s in the calculation below, and the 2 that is carried (from the 25) is also shown.

Now, 5$\times$4 + 2 = 22, so M must be 2. Both Ms are replaced by 2s in the calculation below, and the 2 that is carried is also shown.

Now, 2$\times$4 + 2 = 10, so L must be 0. Both Ls are replaced by 0s in the calculation below, and the 1 that is carried is also shown.

Now, 0$\times$4 + 1 = 1, so K must be 1.

So K, L, M, N, P are 1, 0, 2, 5, 6.

Suppose we are looking for the number KLMNP, and let KLMNP $=x$.

Then the top number, KLMNP4, is KLMNP0 + 4, which is $10x+4$.

The bottom number, 4KLMNP, is 400 000 + KLMNP, which is $400000+x$.

So $(10x+4)\times4=400000+x$. Solving for $x$,$$\begin{align}40x+16&=400000+x\\39x+16&=400000\\39x&=399984\\x&=399984\div39=10256\end{align}$$

So K, L, M, N, P are 1, 0, 2, 5, 6.

You can find more short problems, arranged by curriculum topic, in our short problems collection.