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N000ughty Thoughts

Age 14 to 16 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.