Cut a piece of dough into $3$ even pieces

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How could you cut a piece of dough into $3$ even pieces? Cutting it into $2$ is easy, but it's not that trivial for greater numbers.

If you can cut it into $n$ pieces, you could repeat the process on each piece, getting any multiplication of $n$, so to ask a more general question:

Let $p\in\mathbb{N}$ be a prime number. How can you cut a piece of dough into $p$ pieces?

P.S. I couldn't find a good tag, any bright idea?

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It is straightforward to divide an arbitrary line segment into $n$ equal pieces if one has a unit measure and can construct parallels. Let the segment be $AB$ - take an arbitrary line through $A$ which is not the line $AB$ and mark off $n$ unit intervals so that $AC$ has length $n$ units. Construct $CB$ and parallels to $CB$ through the marked off points - they cut $AB$ evenly into $n$ equal pieces. Arrange your dough in a rectangle and use this construction.