mathschallenge.net logo

Product Of Zero

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 times 5 = 35, 3 times 5 = 15, 1 times 5 = 5
 68:6 times 8 = 48, 4 times 8 = 32, 3 times 2 = 6
 45:4 times 5 = 20, 2 times 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: 2times 5 = 10, 1times 0 = 0.
25, 52=>1045, 54=>2056, 65=>3058, 85=>40

Less obviously 69 needs three steps: 6times 9 = 54, 5times 4 = 20, 2times 0 = 0.
55=>5559, 95=>4569, 96=>5478, 87=>56

Giving 9 + 8 + 7 = 24 solutions.

Which numbers under 1000 have this property?

Problem ID: 37 (Mar 2001)     Difficulty: 1 Star

Only Show Problem