Download Chinese Remainder Theorem

Chinese Remainder Theorem
History of Math
Name:
Chinese Remainder Theorem
Solving Linear Conguences
Multiplication, Addition, Division and Subtraction are all a little different mod(n). We call mod a
homomorphism, and it’s important to understand the structure of the integers modulo n, n . First of all
n
 {0,1, 2,3,.....n  1}
and -3 mod(n) = n-3. Also, the multiplicative inverse of k mod(n) doesn’t necessarily have to exist.
Example: Consider the group
10
2 + 8 = 0 mod(10) means that -2 mod (10 ) = 8
Also, although 3(7) = 1 mod(10), two does not have a multiplicative inverse, because 2(5) = 0 mod(10).
We call 2 a zero divisor mod(10). It is also important to note (a  b) mod(10)  a mod(10)  b mod(10)
and (ab) mod(10)  a mod(10)b mod(10) . Why do you think this is true?
Exercise: Write all the additive and multiplicative inverses in
10
, if any exist.
Answer:
Now let us consider a more complicated problem. What are the solutions to the modular equation
1233x  45  9090 mod(24)
We can still subtract 45 from both sides
1233x  9045 mod(24)
But this means that
1233mod(24) x mod(24)  9045 mod(24)
9 x  21mod(24)
By definition of mod(24), this means that there is an integer k such that
9x  24k  21
We can divide both sides of this equations by 3 to achieve
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number
Chinese Remainder Theorem
History of Math
Name:
3x  8k  7
This means 3x {..., 33  25, 17, 9, 1, 7,15, 23,31,39, 47,55, 63, 71, 79,87, } . The numbers for
which there is a solution are 3x = 15, 39, 63, 87, .....
In which case, the values of x would be {..,-11,-3, 5, 13, 21, 29, ... } = 5  8 .N
Exercises:
1. 1235 x  45  9090 mod(24)
2. 1235 x  45  9090 mod(11)
3. 1235 x  45  9087 mod(11)
4. 1235 x  45  9090 mod(24)
5. Solve the simultaneuos equations
3x  4 mod(7), 4 x  5 mod(8), and 5 x  6 mod(9)
Written by L Marizza A Bailey with problems from the Art of Problem Solving Introduction to Number