FizzBuzz in BASH

#!/bin/bash

for i in {1..100}
do
    let "a = ($i % 3)"
    let "b = ($i % 5)"
    if [ $a -eq 0 ] && [ $b -eq 0 ] ; then
        echo "fizzbuzz"
    else
        if [ $a -eq 0 ] ; then
            echo "fizz"
        else
            if [ $b -eq 0 ] ; then
                echo "buzz"
            else
                echo $i
            fi
        fi
    fi
done