Checking Nakayama's lemma for a specific example

Question

Verify Nakayama's lemma that says if $IM=M$ then $M=0$ by hand in the particular case where we take our ring $R$ to be $\mathbb{Z}$, our module $M$ to be $\mathbb{Z}/16 \mathbb{Z}$, and our ideal $I$ to be $(11)$.
In other words: check that $IM=M$. I cannot see how this makes sense! How is $M=0$ when $M=\mathbb{Z}/16\mathbb{Z}$?

Answer 1

The gcd of 11 and 16 is 1. Thus there exists an x,y $\in \mathbb{Z}$ such that 16x + 11y = 1. If we reduce mod 11, this says that 16x $\equiv$ 1 (mod 11). For our case, if x = 9, then 9*16 = 144 = 13(11) + 1. Thus 144*M = 0, since 144 is a multiple of 16.