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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.