word problem: "consistent" numbers

[sallyk57]

Constance enjoys playing with numbers. Her favorite game is to take a number, work out the product of its digits, and do the same with the number she has obtained – until she ends up with a one-digit number.

23 -> 2 x 3 = 6
54 -> 5 x 4 = 20 à 2 x 0 = 0
999 -> 9 x 9 x 9 = 729 à 7 x 2 x 9 = 126 à 1 x 2 x 6 = 12 à 1 x 2 = 2

The persistence of a number is the number of multiplication steps necessary to get to a one-digit number. For example, the persistence of 6 is 0, the persistence of 23 is 1, for 54 it is 2, and for 999 it is 4. Constance is especially interested in the numbers whose persistence is greater than or equal to 4, and she calls them consistent numbers.

What is the smallest consitent number?

 

[galactus]

I've seen this before. The official name is 'multiplicative persistence'.

Try 77.

1. 7*7=49
2. 4*9=36
3. 3*6=18
4. 1*8=8

4 steps. This is the smallest 4 persistence number, that is, it takes 4 steps to get to a single digit.

This is your goal, isn't it?.

 

[TchrWill]

Constance enjoys playing with numbers. Her favorite game is to take a number, work out the product of its digits, and do the same with the number she has obtained – until she ends up with a one-digit number.

23 -> 2 x 3 = 6
54 -> 5 x 4 = 20 à 2 x 0 = 0
999 -> 9 x 9 x 9 = 729 à 7 x 2 x 9 = 126 à 1 x 2 x 6 = 12 à 1 x 2 = 2

The persistence of a number is the number of multiplication steps necessary to get to a one-digit number. For example, the persistence of 6 is 0, the persistence of 23 is 1, for 54 it is 2, and for 999 it is 4. Constance is especially interested in the numbers whose persistence is greater than or equal to 4, and she calls them consistent numbers.

What is the smallest consitent number?

If I understand you correctly, the smallest consistent number equal to 4 is 14, followed by 27, 39, 41, 93, 139, 193, 319, 391, 913m 931, ....

5 = 15, 35, 53, 135, 153, ....

6 = 16, 23, 32, 61,....