  Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf
representation of the multiplicative inverse of $2$ modulo $F_n$, for every
positive integer $n$ not divisible by $3$, where $F_n$ denotes the $n$th
Fibonacci number. We determine the Zeckendorf representation of the
multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$
and for all positive integers $n$ with $\gcd(a, F_n) = 1$. Our proof makes use
of the so-called base-$\varphi$ expansion of real numbers.
