Challenge Level

Thank you Vassil Vassilev, Yr 11, Lawnswood High School, Leeds for the solution below, well done! Danny Ng, 16, from Milliken Mills High School, Canada sent a very similar solution and so did Koopa Koo of Boston College.

First I tried to convince myself that 100! has 24 noughts. I did that by counting the number of 5s in the numbers from 1 to 100 which are all multiplied together. I did that because a zero at the end can only be produced by multiplying an even number with a 5 and there are more even numbers than multiples of 5 in the product.

Range of numbers | Number of 5 |
---|---|

1 - 10 | 2 |

11 - 20 | 2 |

21 - 30 | 3 |

31 - 40 | 2 |

41 - 50 | 3 |

51 - 60 | 2 |

61 - 70 | 2 |

71 - 80 | 3 |

81 - 90 | 2 |

91 - 100 | 3 |

total | 24 |

Also there is another way to find the number of zeros. This is by:

100 / 5 = 20 | this is the number of multiples of 5 |

20 / 5 = 4 | this is the number of multiples of 5 ^{2} |

When we add this two together we get 24 which is exactly the number of noughts in 100!

So to see if my rule works I will find how many noughts are there in 1000!:

1000 / 5 = 200 | this is the number of multiples of 5 |

200 / 5 = 40 | this is the number of multiples of 5 ^{2} |

40 / 5 = 8 | this is the number of multiples of 5 ^{3} |

8 / 5 = 1.6 | this is the number of multiples of 5 ^{4} . |

The number of zeros has to be a whole number so the number of
multiples of 5 ^{4} is 1 which is the integer part of 1.6
(written [1.6] ). Note that the process stops when division by 5
gives a number less than 5. If we add those answers together we
will get the number of noughts. 200 + 40 + 8 + 1 = 249. From here
we see that my rule works.

So to get the number of noughts in 10 000! we just divide by 5 to get the number of 5s:

10000 / 5 = 2000 | this is the number of multiples of 5 |

2000 / 5 = 400 | this is the number of multiples of 5 ^{2} |

400 / 5 = 80 | this is the number of multiples of 5 ^{3} |

80 / 5 = 16 | this is the number of multiples of 5 ^{4} |

16 / 5 = 3.2 | so [3.2] = 3 is the number of multiples of 5 ^{5} |

2000 + 400 + 80 + 16 + 3 = 2499

To get the number of noughts in 100 000!:

100000 / 5 = 20000 | this is the number of multiples of 5 |

20000 / 5 = 4000 | this is the number of multiples of 5 ^{2} |

4000 / 5 = 800 | this is the number of multiples of 5 ^{3} |

800 / 5 = 160 | this is the number of multiples of 5 ^{4} |

160 / 5 = 32 | this is the number of multiples of 5 ^{5} |

32 / 5 = 6.4 | so [6.4]= 6 is the number of multiples of 5 ^{6} |

6.4 / 5 = 1.28 | so [1.28] = 1 is the number of multiples of 5 ^{7} |

20000 + 4000 + 800 + 160 + 32 + 6 + 1= 24999

Here is how Koopa Koo gave the solution for 1 000 000!.

Let [x] denotes the greatest integer that does not exceed x.

The number of right most zeros of 1 000 000! = [1000000/5]
+[1000000/5 ^{2}] +[1000000/5 ^{3}] +[1000000/5
^{4}] +[1000000/5 ^{5}] +[1000000/5 ^{6}]
+[1000000/5 ^{7}] + [1000000/5 ^{8}] = 200000 +
40000 + 8000 + 1600 + 320 + 64 + 12 + 2 = 249998.