## 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 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?

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