MTH 346: The Chinese Remainder Theorem
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1 MTH 346: The Chinese Remainder Theorem March 3, Introduction In this lab we are studying the Chinese Remainder Theorem. We are going to study how to solve two congruences, find what conditions are needed to ensure a solution, and figure out general formulas to solve these congruences. 1.1 Notation and Definitions In this section, we will define some notation and definitions that will help prove our proofs. Divides: We say a divides b and write a b if there exists an integer c such that b = ac. For example 2 6 since 6 = 2 3 and 3 Z. One way to remember how to write this is that the smaller number goes first. For example, I would not write 6 2. Definition Prime: A prime number is a whole number greater than 1 which is divisible by only itself and 1. Definition of Greatest Common Divisor 1: This is denoted as gcd(a,b) and we say that d is the greatest common divisor of a and b, written d = gcd(a,b) if the following conditions are satisfied: 1. d a and d b. 2. If c a and c b, then c d. Lab 2 research question 3: Suppose that gcd(a, b)= 1 and that x = x 0 and y = y 0 is a solution to ax + by = 1. Then all solutions to this equations are given by x = x 0 + mb and y = y 0 ma, where m is any integer. 1
2 Lab 2 Research Question 5: Let a, b, and k be integers with a or b not equal to 0. Suppose that d = gcd(a, b), and x = x 0 and y = y 0 is a solution to ax + by = d. Then all solutions to the equation ax + by = kd, are given by x = kx 0 + m(b/d) and y = ky 0 m(a/d), where m is any integer. 2 The Chinese Remainder Theorem for Two Congruences Research Question 1 Complete the statement of the Chinese Remainder Theorem for two congruences. 2.1 Example Solve the following pairs of congruences. Suppose x 0 (mod 2) and x 1 (mod 3). We start with x 0 (mod 2) and is equivalent to 2 x. By the definition of divides y Z, such that 2y = x. Plug in x to x 1 (mod 3), 2y 1 (mod 3) 2y 4 (mod 3) y 2 (mod 3) 3 y 2, By the definition of divides, k Z, such that 3k = y 2.Solve for y and y = 3k + 2. Plug y in to 2y = x, therefore 2(3k + 2) = x Thus, we have found a solution. 2.2 Answer 6k + 4 = x 6k = x 4 6 x 4 x 4 (mod 6). Theorem: If m 1 and m 2 are positive integers such that m 1 and m 2 are relatively prime, then for any integers a 1 and a 2, the pair of x a 1 (mod m 1 ) and x a 2 (mod m 2 ) 2
3 has a unique solution x modulo m 1 m 2. Proof: Suppose x a 1 (mod m 1 ) and x a 2 mod m 2. Let x a 1 (mod m 1 ) m 1 x a 1 by the definition of congruences. By the definition of divides y Z, such that m 1 y = x a 1. We solve for x, m 1 y + a 1 = x. We substitute x in to our x a 2 (mod m 2 ) equation thus m 1 y + a 1 a 2 (mod m 2 ). We subtract a 1 and m 1 y a 2 a 1 (mod m 2 ). This is equivalent to m 2 a 2 a 1 m 1 y by the definition of congruences. By the definition of divides r Z, such that m 2 r = a 2 a 1 m 1 y. we add m 1 y to both sides and get m 2 r + m 1 y = a 2 a 1. This is similar to the form of rx + sy = t which has a solution by Lab 2 Research Question 3, for all a 2 a 1, if the gcd(m 1, m 2 )=1. Furthermore, by Lab 2 Research Question 5, y = ky 0 + m 2 l where l Z. We plug y into the equation m 1 y + a 1 = x, such that m 1 (ky 0 + m 2 l) + a 1 = x We distribute m 1 to (ky 0 + m 2 l) + a 1 = x such that m 1 ky 0 + m 1 m 2 l + a 1 = x. We (mod m 1 m 2 ) to both sides m 1 ky 0 + m 1 m 2 l + a 1 x (mod m 1 m 2 ) m 1 ky 0 + a 1 x (mod m 1 m 2 ). Thus, we have shown that when the gcd(m 1, m 2 ) a 2 a 1, then m 1 ky 0 + a 1 (mod m 1 m 2 ). x 3 A More General Theorem Research Question 2 For what values of a 1, a 2, m 1, and m 2 will the pair of congruences x a 1 (mod m 1 ) and x a 2 (mod m 2 ) have a solution modulo m 1 m 2? If there is a solution x 0 (mod m 1 m 2 ), find the form of all other solutions x (mod m 1 m 2 ) in terms of x Answer Theorem: The values of a 1, a 2, m 1, and m 2 need to satisfy the condition of gcd(m 1, m 2 ) a 2 a 1, so that the pair of congruences x a 1 (mod m 1 ) and x a 2 (mod m 2 ) have a solution modulo m 1 m 2. The form of all other solutions x (mod m 1 m 2 ) in terms of x 0 is m 1 ky 0 + (m 1 m 2 /d)l x (mod m 1 m 2 ) where x, y 0, m 1, m 2, d, l Z and d = gcd(m 1, m 2 ). Proof: Let x a 1 (mod m 1 ), x a 2 (mod m 2 ) and d a 2 a 1, where d = gcd(m 1, m 2 ). By the definition of congruences x a 1 (mod m 1 ) can be written iff m 1 x a 1. y Z 3
4 iff m 1 y = x a 1. Solve for x, x = m 1 y + a 1. Plug x = m 1 y + a 1 into x a 2 (mod m 2 ), thus m 1 y + a 1 a 2 (mod m 2 ). Subtract a 1 from both sides, iff m 1 y a 2 a 1 (mod m 2 ). By the definition of congruences, m 1 y a 2 a 1 (mod m 2 ) iff m 2 a 2 a 1 m 1 y. r Z iff m 2 r = a 2 a 1 m 1 y by the definition of divides. Add m 1 y to both sides, iff m 2 r + m 1 y = a 2 a 1. By lab 2 Research question 5, note that m 2 r + m 1 y = a 2 a 1 is in the form ax + by = c which tells us that it has a solution when d a 2 a 1. Therefore, y = ky 0 + (m 2 /d)l, l Z. Plug in y, so that Then we (modm 1 m 2 )to both sides so that, m 1 y + a 1 = x iff (ky 0 + (m 2 /d)l) = x a 1 iff m 1 ky 0 + (m 1 m 2 /d)l + m 1 a 1 = x. (m 1 ky 0 + (m 1 m 2 /d)l + m 1 a 1 )(modm 1 m 2 ) = x(modm 1 m 2 ) iff m 1 ky 0 + (m 1 m 2 /d)l + m 1 a 1 x (mod m 1 m 2 ). Thus, we have shown the form of all other solutions x (mod m 1 m 2 ) in terms of x 0. 4 Solving Lots of Congruences Research Question 3 Give a statement of the Chinese Remainder Theorem (as in Research Question 1) for n congruences. 4.1 Answer Theorem: If m 1, m 2, m 3,..., m k are positive integer such that they are pairwise relatively prime, then for any integers a 1, a 2, a 3,..., a k the pair of congruences x a 1 (mod m 1 ) x a 2 (mod m 2 ) x a 3 (mod m 3 ). x a k (mod m k ) has a unique solution x modulo M, where M = m 1 m 2 m 3... m k. Proof: Suppose m, a, x Z and m 1, m 2, m 3,..., m k are positive integer such that they are pairwise relatively prime so that for any integers a 1, a 2, a 3,..., a k the pair of congruences x a 1 (mod m 1 ) 4
5 x a 2 (mod m 2 ) x a 3 (mod m 3 ). x a k (mod m k ). By Research Question 1 if we take two pairs of congruences x a 1 (mod m 1 ) and x a 2 (mod m 2 ), and we know they are relatively prime then we have a unique solution so that, x b 1 (mod m 1 m 2 ). Then we do the same thing to the next congruence pairs with and get x b 1 (mod m 1 m 2 ) and x a 3 (modm 3 ) x b 2 (mod m 1 m 2 m 3 ), because they are relatively prime. We continue this until we get our unique solution, x b k 1 (mod m 1 m 2 m 3...m k ). Thus we have shown that if m 1, m 2, m 3,..., m k are pairwise relatively prime then the pair of congruences has a unique solution x modulo M, where M = m 1 m 2 m 3... m k. 5 Explicit Formulas Research Question 4 With the assumptions of Research Question 1, find formulas for c 1 and c 2 so that x = c 1 m 1 + c 2 m 2, is a solution to the pair of congruences in Research Question Answer Theorem: Assume Research Question 1 then c 1 and c 2 in the form x = a 2 m 1 m 1 is a solution to the pair of congruences in Research Question 1. 5
6 Proof: Consider x (mod m 1 ) and x (mod m 2 ). Since x = a 2 m 1 m 1 1 (mod m 2 )+a 1 m 2 m 1 2 (mod m 1 ) we can plug x in to x (mod m 1 ), We distribute (mod m 1 ), (a 2 m 1 m 1 2 ) (mod m 1 ). a 2 m 1 m 1 1 (mod m 2 ) (mod m 1 ) + a 1 m 2 m a 1 1 (mod m 1 ), since the first term has m 1 and we (mod m 1 ) we know that a 2 m 1 m 1 1 (mod m 2 ) (mod m 1 ) is 0. Also, the second term equals 1, because we multiplied by its inverse. Thus a 1 (mod m 1 ). We can also show this with the congruence x (mod m 2 ). Plug (a 2 m 1 m 1 1 (mod m 2 ) + a 1 m 2 m 1 2 ) (mod m 1 ) for x, We distribute (mod m 2 ), (a 2 m 1 m 1 ) (mod m 2 ). a 2 m 1 m 1 (mod m 2 ) a (mod m 2 ). Since the first term is being multiplied by its inverse it equals 1 and the second term is 0, because we (mod m 2 ). Thus a 2 (mod m 2 ). Therefore we have shown that x = a 2 m 1 m 1 is a solution to the pair of congruences x a 1 (mod m 1 ) and x a 2 (mod m 2 ). 6 System of Congruences Research Question 5 With the assumptions of Research Question 3, find a formula for x so that x will be a solution to the system of congruences x a i mod(m i ). 6
7 6.1 Answer Theorem: We assume Research Question 3.The formula x = k i=1 a i m i m 1 i (mod m i ) where M i = m 1,m 2,m 3,...,m k m i, k i,is a solution, x, to the system of congruences x a i (mod m i ). Proof: WLOG: Assume x a i (mod m i ) and let x = a 1 m 1 m 1 1 (modm 1 )+a 2 m 2 m 1 2 (modm 2 )+a 3 m 3 m 1 3 (modm 3 )+...+a k m k m 1 (modm k). We plug in this value of x for x a i (mod m i ), so that, (a 1 m 1 m 1 1 (modm 1 )+a 2 m 2 m 1 2 (modm 2 )+a 3 m 3 m 1 3 (modm 3 )+...+a k m k m 1 k (modm k)) a i (mod m i ) Note that m i m j i j and m j 0 (mod m i ), i j, such that a i (M i M 1 i (modm i )) a i 1 a i (modm i ). Therefore we have shown that this formula works and x will be a solution to the system of congruences x a i (mod m i ). k 7 Conclusion In this lab we have studied the Chinese Remainder Theorem and have come to some conditions. In order to have a unique solution for a pair of congruences then m 1 and m 2 need to be relatively prime. Also, if the gcd(m 1, m 2 ) a 2 a 1, then m 1 ky 0 + (m 1 m 2 /d)l + m 1 a 1 x (mod m 1 m 2 ) is the general form. We figured out a formula, x = a 2 m 1 m 1 which is a solution to a pair of congruences. For a system of congruences, they need to be pairwise relatively prime to have a unique solution and if this is so, then by this formula, x = k i=1 a i m i m 1 i (mod m i ) we can find a solution. After all this research we understand how to apply the Chinese Remainder Theorem and what conditions apply. 7
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