Chinese Remainder Theorem:
Ask someone to pick a number from 1 to 1000. they must tell you the remainder that they get when they divide their number by first 7, then 11, then 13. You then have enough information to tell what their original number was. Here is how to do it.
Let's say the remainders after dividing by 7, 11, and 13 respectively were a, b, and c. Compute the expression:
715a + 364b + 924c
This number has the desired properties but it is generally larger than 1000. To determine your friend's number, keep subracting 1001 from this result until you get a number less than 1000.
Example:
Let's say your friend's number were 200.
Then a=4, b=2, and c=5
Therefore: 715(4) + 364(2) + 924(5) = 8208
If you subtract 8 groups of 1001, then you get,
8208  8008 = 200
Your problem is to figure out how this trick works. The answer
lies in the following tables of modular arithmetic mod 7, mod 11, and mod
13.
143 = 11x13
Multiples of 143 
Value
Mod 7 
91 = 7x13
Multiples of 91 
Value
Mod 11 
77 = 7x11
Multiples of 77 
Value
Mod 13 
143
286 429 572 715 858 1001

3
6 2 5 1 4 0

91
182 273 364 455 546 637 728 819 910 1001

3
6 9 1 4 7 10 2 5 8 0 
77
154 231 308 385 462 539 616 693 770 847 924 1001 
12
11 10 9 8 7 6 5 4 3 2 1 0 
Home  Academic
 Personal  Research
 Site Map  Search
Web  Email
 Links