# Sum product of digits

Question: Given any whole number take the sum of the digits, and the product of the digits, and multiply these together to get a new whole number.
For example, starting with 6712, the sum of the digits is (6+7+1+2) = 16, and the product of the digits is (6*7*1*2) = 84. The answer in this case is then 84 x 16 = 1344.
If we do this again starting from 1344, we get (1+3+4+4) * (1*3*4*4) = 576
And yet again (5+7+6) * (5*7*6) = 3780
At this stage we know what the next answer will be (without working it out) because, as one digit is 0, the product of the digits will be 0, and hence the answer will also be 0.
Can you find any numbers to which when we apply the above mentioned rule repeatedly, we never end up at 0?

Three such numbers are 1, 135 and 144.
It seems that most numbers will eventually end up at 0 when we apply the rule repeatedly.
But there are a few numbers that have the property that when we apply the rule repeatedly, we never end up at 0.
Start with 332, then we get (3+3+2) * (3*3*2) = 144
And then (1+4+4) * (1*4*4) = 144
Thus if we reach 144, we stay there however many times we apply this rule. We say that 144 is fixed by this rule. Now try 233 or 98 or 332 or 1224. They all fall into the same group i.e. we reach 144.
There is another number that is fixed by this rule; it is 1 (because the sum of the digits
of 1 is 1, and the product of the digits is 1 so, starting with 1, the answer is 1 * 1 = 1).
And the third one is 135.