Chapter 4 Complementary Sets Of Systems Of Congruences Proceedings NCUR VII. è1993è, Vol. II, pp. 793í796. Jeærey F. Gold Department of Mathematics, Department of Physics University of Utah Don H. Tucker Department of Mathematics University of Utah Introduction We introduce remodulization and use it to characterize the complementary sets of systems of congruences. The following is an excerpt of a continuing eæort to characterize systems of congruences. Remodulization Deænition 1 If a and b are integers, then a mod b = fa; a æ b; a æ 2b;:::g : We will write x ç a mod b, meaning that x is an element oftheseta mod b. èthe symbol ç is not an equality symbol, but rather a specialized form of the ë2" or ëis an element of:::" symbolè. The common terminology is to say that x 1

CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 2 is congruent to a modulo b. These sets are also frequently called residue classes since they consist of those integers which, upon division by b, leave a remainder èresidueè of a. It is customary to write a as the least non-negative residue. Deænition 2 If a 1 ;a 2 ;:::;a n,b 2 Z, then ëa 1 ;a 2 ;:::;a n ëmodb =èa 1 mod bèëèa 2 mod bèëæææëèa n mod bè = në a i mod b: Theorem 1 èremodulization Theoremè Suppose a, b, and c 2 Z and c 0, then a mod b =ëa; a + b;:::;a+ bèc, 1èë mod cb : Proof. We write a mod b = f ::: a, cb; a, èc, 1èb; ::: a, b; a; a + b; ::: a +èc, 1èb; a + cb; a +èc +1èb; ::: a +è2c, 1èb; ::: g and upon rewriting the columns, a mod b = f ::: a, cb; a + b, cb; ::: a +èc, 1èb, cb; a; a + b; ::: a +èc, 1èb; a + cb; a + b + cb; ::: a +èc, 1èb + cb; ::: g : Then, forming unions on the extended columns, the result follows. We refer to this process as remodulization by a factor c. Complementary Sets Using Deænition 2, the complementary set of amodbis given by ëa mod bë c = Z nfa mod bg =ë0; 1; 2;:::;a, 1;a+1;:::;b, 1ë mod b: In this case, the complementary set consists of b, 1 congruences modulo b. We will always refer to the size of a complementary set with respect to a speciæc modulus. The following represents a system of congruences: : x ç a 1 mod b 1 x ç a 2 mod b 2. x ç a n mod b n è4.1è

CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 3 Characterizing the size of the complementary set of a system of congruences è4.1è is equivalent to counting those integers which are not elements of any of the given congruences. Our method will be to remove the set of numbers satisfying the system of congruences è4.1è from the set of integers Z in a systematic way. Note that Z can be written as a complete residue class, ë1; 2;:::;bë mod b, for all b1. All numbers satisfying the ærst congruence in è4.1è will be removed from Z, leaving the complementary set for that congruence. Then we iterate the process by removing all integers satisfying the second congruence from this remaining set, and so on. Stated another way, we are interested in determining the size of ëa 1 mod b 1 ë a 2 mod b 2 ëæææëa n mod b n ë c = Z n è në a i mod b i è where the b i are pairwise relatively prime. Our method is to determine the number of èremainingè congruences needed to characterize this complementary set. Using the remodulization method, the set of congruences can be expressed in terms of the common modulus, Q n mod b i and the integers, Z, can be expressed in terms of a complete residue class of the same modulus. To illustrate this procedure, suppose we have the following system of congruences ç x ç a1 mod b 1 ; x ç a 2 mod b 2 where gcdèb 1 ;b 2 è = 1. By the Chinese Remainder Theorem ë1,2ë, these intersect in a unique residue class modulo b 1 b 2. Remodulizing the congruences by b 2 and b 1, respectively, this system can be expressed as ç a1 mod b 1 = ëa 1 ;a 1 + b 1 ;:::;a 1 + b 1 èb 2, 1èë mod b 1 b 2 a 2 mod b 2 = ëa 2 ;a 2 + b 2 ;:::;a 2 + b 2 èb 1, 1èë mod b 1 b 2 The ærst congruence, after remodulization, consists of b 2 congruences mod b 1 b 2 while the second remodulized congruence consists of b 1 congruences mod b 1 b 2 ; furthermore, Z consists of b 1 b 2 congruences èmodulo b 1 b 2 è. Therefore, subtracting b 1 and b 2 from b 1 b 2 and adding one to this sum èthe unique intersection of the two congruences was removed twice from b 1 b 2 è, we obtain b 1 b 2, b 1, b 2 +1 remaining congruences mod b 1 b 2 in the complementary set ëa 1 mod b 1 ë a 2 mod b 2 ë c :

CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 4 However, b 1 b 2, b 1, b 2 + 1 can be rewritten as èb 1, 1èèb 2, 1è; this is typical. S n Theorem 2 The complementary set of f a i mod b i g,wheretheb i are pair- b i. wise relatively prime, contains exactly Q n èb i,1è congruences modulo Q n Proof. Suppose we have asystem of congruences è4.1è where the b i are pairwise relatively prime. We have already found that the complementary sets of a 1 mod b 1 and a 1 mod b 1 ë a 2 mod b 2 consist of èb 1,1è congruences modulo b 1 and èb 2, 1èèb 1, 1è congruences modulo b 1 b 2, respectively. For the induction argument, suppose we have found the complementary set up to k th congruence of è4.1è to consist of the union of Q k èb i, 1è congruences Q k modulo b i; then we remove from it the set of numbers congruent fa k+1 mod b k+1 g. The complementary set of the latter congruence is comprised of èb k+1, 1è congruences modulo b k+1. Each of these congruences shares a unique intersection with each of the congruences in the remaining complementary set; there are èb k+1, 1è Q k èb i, 1è such intersections Q k+1 modulo b i. Therefore, the remaining complementary set, after removing the èk +1è st congruence from the remaining complementary set, consists of congruences modulo k+1 Y b i. k+1 Y èb i, 1è If the moduli b i are primes, then we may use Euler's phi function ë3,4ë, or totient, to formulate the complementary set of a system of congruences. The totient æèmè counts the number of integers not exceeding m which are relatively prime to m. For any prime p, æèpè =p, 1; moreover, because the totient is multiplicative, æèp 1 p 2 æææp n è=æèp 1 èæèp 2 è ææææèp n è= Q n èp i, 1è. Corollary 1 Suppose we have a system of congruences where themoduli p i are primes and p i 6= p j,fori 6= j. Then the complementary set of consists of æ è ny p i! : x ç a 1 mod p 1 x ç a 2 mod p 2. x ç a n mod p n congruences modulo ny p i.

CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 5 Deænition 3 The density of the complementary set of a system of congruences è4.1è, where gcdèb i ;b j è=1for i 6= j, with respect to the set Z, is ny ç ç bi, 1 çènè = : As an illustration, we calculate the size and density of the complementary set of the following system: : b i x ç 1mod3 x ç 2mod5 x ç 3mod7 The complement of the ærst congruence, 1 mod 3, is ë2; 3ë mod 3, a union of two congruences modulo 3. The complement of 2 mod 5 is ë1; 3; 4; 5ë mod 5. Each of the congruences in the complementary set modulo 3 shares a unique intersection with the congruences of the complementary set modulo 5; there are è3, 1èè5, 1è = remaining congruences modulo 15. Finally, removing all numbers satisfying 3 mod 7 from these remaining congruences, the complementary set consists of è3, 1èè5, 1èè7, 1è = 4 congruences modulo 105. The density of the complementary set with respect to the set Z at each step in the process is 2=3, =15, and 4=105, respectively. If a system consists of æ bi congruences of the same modulus b i, for each b i, we have the following extension of Theorem 2. Suppose : x ç ëa 1;1 ;a 1;2 ;:::;a 1;æb 1 ëmodb 1 x ç ëa 2;1 ;a 2;2 ;:::;a 2;æb 2 ëmodb 2. x ç ëa n;1 ;a n;2 ;:::;a n;æbn ëmodb n è4.2è where the b i are pairwise relatively prime, and a i;j 6= a i;k for all j 6= k, and æ bi b i. The complementary set of the ærst congruence is the union of b 1, æ b1 congruences. Likewise, the complementary set of the second congruence contains b 2,æ b2 congruences; their intersection contains èb 2,æ b2 èèb 1,æ b1 è congruences modulo b 1 b 2. Iterating the process, the complementary set of è4.2è consists of Q n èb i, æ bi è congruences modulo Q n b i. At eachstep,however, it is necessary to insure that b i, æ bi 0; otherwise, if æ bi = b i, the complementary set vanishes altogether, since for that particular value of i, ëa i;1 ;a i;2 ;:::;a i;æbi ëisa complete residue class modulo b i, i.e., the entire set Z. n Sn Theorem 3 The complementary set of ëa i;1;:::;a i;æbi ëmodb i o, where a i;j 6= a i;k for j 6= k, and æ bi b i, and the b i are pairwise relatively prime,

CHAPTER 4. COMP. SETS OF SYSTEMS OF CONGRUENCES 6 Q n contains exactly èb i, æ bi è Q n congruences modulo b i. The density of the complementary set is çènè = Q ç n b i,æ bi b i ç. References ë1ë Gold, Jeærey F. and Don H. Tucker, Remodulization of Congruences and Its Applications. To be submitted. ë2ë Gold, Jeærey F. and Don H. Tucker, Remodulization of Congruences, Proceedings - National Conference on Undergraduate Research, èuniversity of North Carolina Press, Asheville, North Carolina, 1992è, Vol. II, pp. 1036í41. ë3ë David M. Burton, Elementary Number Theory èwm. C. Brown Publishers, Iowa, 199è, Second Edition, pp. 156í160. ë4ë Oystein Ore, Number Theory and Its History èdover Publications, Inc., New York, 19è, pp. 109í115.