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There were lots of interesting patterns that could be identified and explained in this problem. Ngoc from Nguyen Truong To junior high school (Vietnam) identified some interesting patterns along with giving the answers. Well done Ngoc. It is also good, once you have identified patterns, to try to explain them in terms of the mathematics they came from. So where do the patterns Ngoc and Andrei (School 205, Bucharest) come from?
Ngoc's contribution:
The highest power of 2 that divides exactly to 100! is 97
The highest power of 3 that divides exactly to 100! is 48
General form:
with all n, k are positive integers, n > k
The highest power of k (called y) that divides exactly to n!
is:
Andrei's Contribution
I started working with 100!.
To calculate the total highest power of two that divides exactly into 100! is calculating by multiplying the corresponding power of two with the number of solutions and multiplying the results, because it is 100!:
(2 6 ) 1 * (2 5 ) 2 * (2 4 ) 3 * (2 3 ) 6 * (2 2 ) 13 * (2 1 ) 25 = 2 6 * 2 10 * 2 12 * 2 18 * 2 26 * 2 25 = 2 97
I use the same method for dividing 100! into powers of three:
Power of 3 | 3n < 100! | Combinations | Result |
3 4 | 1 | 1 | 1 |
3 3 | 3 | 3 - 1 | 2 |
3 2 | 11 | 11 - 3 | 8 |
3 | 33 | 33 - 11 | 2 |
Here the greatest power of three that divides exactly into 100!:
(3 4 ) 1 * (3 3 ) 2 * (3 2 ) 8 * (3 1 ) 22 = 3 4 * 3 6 * 3 16 * 3 22 = 3 48
Now, the general method with n! as the number, and x as the number whose powers enter n!:
I would like to mention that I used for 100! and for the powers of 2 the traditional method, it means 100 is divisible by 2 2 , 98 by 2, ? and so on, and I obtained the same result as before.