For example 102 is divisible by 3.

This quotient is not Harshad because 34 is not divisible by 7.

108 is a Multiple Harshad Number because the process ends at 1:

108/9=12; 12/3=4; 4/4=1.

Find the Multiple Harshad Numbers below 1000.

Hard bonus: Apparently there are only 15095 of these numbers. Can you prove the list is finite?