mathschallenge.net

Problem

Take any 2-digit number and multiply the digits together. If this process is continued, all 2-digit numbers will become a single digit number.

For example,	75:	7 5 = 35, 3 5 = 15, 1 5 = 5
	68:	6 8 = 48, 4 8 = 32, 3 2 = 6
	45:	4 5 = 20, 2 0 = 0

How many 2-digit numbers will finish on zero?

Solution

Obviously any number ending with 0 will give zero directly.

10, 20, 30, 40, 50, 60, 70, 80 and 90
But indirectly, numbers like 25 require two steps: 2 5 = 10, 1 0 = 0.
25, 52=>10	45, 54=>20	56, 65=>30	58, 85=>40
Less obviously 69 needs three steps: 6 9 = 54, 5 4 = 20, 2 0 = 0.
55=>55	59, 95=>45	69, 96=>54	78, 87=>56

Giving 9 + 8 + 7 = 24 solutions.

Which numbers under 1000 have this property?