# Montgomery Reduction - what should the choice of R be?

When we need to compute $$z = xy \text{ mod } N$$ and the Montgomery Reduction of $$x$$ is $$xR^{-1}$$ why should the choice of R be $$2^l$$ where $$l$$ is the length of $$N$$ to the base $$2$$? Why cannot we have a larger $$R$$?

Theoretically $$R$$ needs to be such that $$2^{l-1} < N < R = 2^l$$. But practically $$l$$ is chosen to be multiples of CPU word so if 1 word is 32 bits and $$l$$ comes out to be 30 as per above equation, l will instead be chosen as 32 and then $$R = 2^{32}$$.
$$2^l$$ is chosen because in computers, it is efficient to do operations like multiplication, division, modulus with powers of 2.
Multiplication by $$2^l$$ is equivalent to shifting first $$l$$ bits to the left (and put zero at the place of shifted bits).
Division by $$2^l$$ is equivalent to shifting first $$l$$ bits to the right.
Taking modulo $$2^l$$ is equivalent to choosing the first $$l$$ bits only.