The Chinese Remainder Theorem

1. The Chinese Remainder Theorem

2. Setting the Scene(the problem) • Imagine that you are a commander in the army. • You have lost a major battle and are attempting to regroup. • Before you can do this, you must count the survivors. Originally you have 500 soldiers. • You are a busy commander though and this process would be too strenuous to attempt by hand.

3. An Attempted Solution • Being a wise leader, you realize that you could group the remaining infantry into a nxm matrix. Then, the total solders would just equal n * m. • But, annoyingly enough, a large number of solders remain outside your matrix with no place to stand.

4. An Observation • No matter what size matrix you have, there seems to be people left over. • Although there might be a square solution, finding the correct n and m is difficult.

5. Guessing Not being one to admit defeat, you decide to try different sized groupings anyways: • At first, you decide to group the survivors into groups of 3. After dividing the whole of the army, 1 person had no place in a group. • Then, you decide that groups of 7. Once this was done, 3 people had been left out. • In desperation, you decide to group the army into groups of 11. This left 2 people over.

6. Another Observation • After thinking for awhile, you realize that the total number of solders is going to be some x such that: x = 1 (mod 3) x = 3 (mod 7) x = 2 (mod 11) • Invigorated by these equations, but frustrated with no solution, you seek advice.

7. Advice • Your advisors remind you of the Chinese Remainder Theorem: If m1, m2, ...mk are positive integers with a greatest common divisor of one, then this system: x = a1  (mod m1) x = a2  (mod m2) … x =ak (modmk) Is solved by the modulo m= m1 * m2 * ... *mk

8. Proof • Unconvinced you demand an example of this theorem. It is presented as follows: with our numbers: x = 1 (mod 3) x = 3 (mod 7) x = 2 (mod 11) Now take Mk , with m divided by mk for each k = 1, 2, ... N • What? The top aid explains slowly… Simply multiply all m except for mkto find Mk . Using our numbers: M1 = 7 * 11 = 77 M2 = 3 * 11 = 33 M3 = 3 * 7 = 21

9. Furthermore • Next, we have to find the solutions xk for each of the equationsMk * xk = 1  (modmk). M1 = 7 * 11 = 77 M2 = 3 * 11 = 33 M3 = 3 * 7 = 21 x = 1 (mod 3) x = 3 (mod 7) x = 2 (mod 11) • For our problem, this is: • 77 * x1 = 2 * x1 = 1 (mod 3) -> 77 mod 3 = 2 • 33 * x2 = 5 * x2 =1 (mod 7) -> 33 mod 7 = 5 • 21 * x3 =10 * x3 =1 (mod 11) -> 21 mod 11 = 10 So, we have x1 = 2 x2 = 3 x3 = 10

10. Almost there… • s = a1 * M1 * x1 + a2 * M2 * x2 + ...+ ar * Mr * xr • s = a1 * M1 * x1 + a2 * M2 * x2 + ...+ ar * Mr * xr • s = 1 * 77 * 2 + 1 * 33 * 3 + 1 * 21 * 10 • s = 463 = x (mod 231)

11. Solution • Taking s = 463 = x (mod 231) • Originally have 500 soldiers Solving for x: x = 463 mod 231 = 1 so: s = 1 (mod 231) s = 231*k + 1 k = 0 -> 1 soldier left k = 1 -> 232 soldiers left k = 2 -> 694 soldiers left (too large) • We know now that we either have 1 or 232 soldiers left. • Close enough to easily guess. • Good news is that we have 232 soldiers.

12. Homework • I mailed 200 invitations to this rave. I danced in every song… • We did a dance where we paired off into groups of 5, but one group only had 4… • Then, we had a line dance with 9 to a line, but three lines ended up with 10… • At the end of the evening, everyone left in pairs, except for me… • How many people (including me) attended the event (assuming no uninvited guests showed up)?