Modulo Multiplication Distributes over Modulo Addition

Theorem

Multiplication modulo $m$ is distributive over addition modulo $m$:

$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \Z_m$:

$\left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m +_m \left[\!\left[{z}\right]\!\right]_m}\right) = \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) +_m \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)$

$\left({\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m = \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right) +_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)$

where $\Z_m$ is the set of integers modulo $m$.

That is, $\forall x, y, z, m \in \Z$:

• $x \left({y + z}\right) \equiv x y + x z \pmod m$
• $\left({x + y}\right) z \equiv x z + y z \pmod m$

Proof

Follows directly from the definition of multiplication modulo $m$ and addition modulo $m$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m +_m \left[\!\left[{z}\right]\!\right]_m}\right)\)	\(=\)	\(\displaystyle \left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y + z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x \left({y + z}\right)}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left({x y}\right) + \left({x z}\right)}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x y}\right]\!\right]_m +_m \left[\!\left[{x z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) +_m \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

And the second is like it, namely this:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m\)	\(=\)	\(\displaystyle \left[\!\left[{x + y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left({x + y}\right) z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left({x z}\right) + \left({y z}\right)}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{x z}\right]\!\right]_m +_m \left[\!\left[{y z}\right]\!\right]_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right) +_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19.1$