## 27 Jun 2013

### The Productivity of Numbers: Upper Secondary Mathematics Competition Question

In the previous question, we looked at the digital addition of natural numbers and introduced the concepts of digital sum, digital root and ‘additude’. In this question, we’re going to look at the multiplicative versions of the same concepts.

Take any natural number and multiply together its digits; this gives us its digital product (dp). Repeat this process until we reach a single-digit number; this is its multiplicative digital root (which is a mouthful, so abbreviated to mdr). The number of iterations needed to reach this mdr, we shall call its ‘productivity’.

For example, let’s take the number 3456, then the dp(3456) = 3x4x5x6 = 360. Then, the dp(360) = 0, giving us the mdr(3456) = 0 and the productivity is equal to 2. As a reminder, the additive digital sums form the sequence (3456, 18, 9) so that the additive digital root is equal to 9 and the additude is 2.

Find all those natural numbers below 1000 that have the multiplicative digital root, the productivity, the additive digital root and the additude, all equal to 2.