What is this function?

Number Theory Level 5

( ∏ k = 1 13 k k k ! ) m o d 28752 \large{\left( \prod _{ k=1 }^{ 13 }{ \frac { { k }^{ k } }{ k! } } \right) \mod{28752}}


The answer is 7872.

Department 8 by Department 8 , 27, Israel

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2 solutions

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Lu Chee Ket
Nov 17, 2015

1 / 1

2 2 / 2

3 3 3 / 3 2

4 4 4 4 / 4 3 2

5 5 5 5 5 / 5 4 3 2

6 6 6 6 6 6 / 6 5 4 3 2

7 7 7 7 7 7 7 / 7 6 5 4 3 2

8 8 8 8 8 8 8 8 / 8 7 6 5 4 3 2

9 9 9 9 9 9 9 9 9 / 9 8 7 6 5 4 3 2

10 10 10 10 10 10 10 10 10 10 / 10 9 8 7 6 5 4 3 2

11 11 11 11 11 11 11 11 11 11 11 / 11 10 9 8 7 6 5 4 3 2

12 12 12 12 12 12 12 12 12 12 12 12 / 12 11 10 9 8 7 6 5 4 3 2

13 13 13 13 13 13 13 13 13 13 13 13 13 / 13 12 11 10 9 8 7 6 5 4 3 2

Divided by 2 2 2 2 3 599

Converted into 

1
2
3
4
5
6
 
                            10  10          
        11  11  11  11  11  11  11  11      
    3   3   3   3   3   3   2   2   6   6   
13  13  13  13  13  13  13  13  13  13  13  13

Divided by 2    2   2   2   3   599

209706417310526095716965894400 MOD 28752 = 7872

Note that (4368883693969293660770122800 x 2^4 x 3) MOD (2^4 x 3 x 599) = 7872 but 4368883693969293660770122800 MOD 599 = 164

164 x 2^4 x 3 = 7872

0 Helpful 0 Interesting 0 Brilliant 0 Confused

This solution made no sense.

Pi Han Goh - 5 years, 5 months ago

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please post your solution or give a hint

Dev Sharma - 5 years, 5 months ago

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Factor LHS and RHS completely, apply Chinese Remainder Theorem . It's a tedious process.

Pi Han Goh - 5 years, 5 months ago

Please learn latex it will help a lot of people.

Department 8 - 5 years, 6 months ago

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