num=12
((num++))       # This just does the arithmetic expansion, which changes num, so has an effect.
echo "num is $num"

((num % 3))     # This just does the arithmetic expansion, which does NOT change num, so has no effect.
echo "num is $num"

echo mod: $((num % 3))  # what does this do? What if you leave out the $?

num2=$((num++)) # This substitutes the result, so sets num2 and changes num
echo "num is $num, num2 is $num2"

num2=$((num % 3)) # This substitutes the result, so sets num2, but does not change num
echo "num is $num, num2 is $num2"