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

Working it out by doing the calculation

4$\times$4 = 16, so P must be 6

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6$\times$4 + 1 = 25, so N must be 5

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5$\times$4 + 2 = 22, so M must be 2

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2$\times$4 + 2 = 10, so L must be 0

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

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

Using algebra

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.