Modular Arithmetic Basics (1) The “floor" function is defined by the formula bxc := ( the greatest integer less than or equal to x): This is also known as \the greatest integer function," and in old texts is denoted by (whole) brackets. Examples: b3:789c = 3; b−3:789c = −4. (2) The \greatest common divisor," abbreviated gcd, of a set of integers is, of course, the largest positive integer that divides every integer in the set. Examples: gcd(24; 52) = 4; gcd(54; 42) = 6. (3) The \mod" operator is defined as follows: x mod m := x − m · bx=mc if m 6= 0. For positive integers x and m, x mod m = the remainder in integer division of x by m. Examples: 110 mod 26 = 6; −52 mod 26 = 0. (4) The \mod" relation is defined as follows: a ≡ b (mod m) if and only if a mod m = b mod m: The above definitions make sense even for real numbers. When a; b; m are integers and m > 0, a ≡ b (mod m) if and only if a − b is a multiple of m: Examples: 110 ≡ 6 (mod 26); −80 ≡ 24 (mod 26). (5) Modular arithmetic and algebra behave \as expected" for the operations of addition, subtraction, and multiplication. If a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m), a − c ≡ b − d (mod m), and ac ≡ bd (mod m). (6) But division and cancellation are trickier. Here are the cancellation rules. ad ≡ bd (mod m) if and only if a ≡ b (mod m); assuming gcd(d; m) = 1: ad ≡ bd (mod m0d) if and only if a ≡ b (mod m0); assuming d 6= 0: Combined: ad ≡ bd (mod m) if and only if a ≡ b (mod m= gcd(d; m)): (7) Let a; x; m be integers with m > 0. Let g = gcd(a; m). The number of solutions of ax ≡ b (mod m) in the set f1; 2; : : : ; mg is 0 if g does not divide b and is g if gjb, and then if x0 is one solution, then all solutions are given by x = x0 + (m=g)k for k = :::; −2; −1; 0; 1; 2; 3;:::, and g of them are in a complete set of residues. (8) Let a; b be integers. Then ajb, read \a divides b," if and only if b is a multiple, i.e., an integer multiple, of a: b = ka for some integer k. Examples: 7j98; −5j100; but 8 does not divide 26. Graham, Knuth, and Patashnik's divisibility notation: anb if and only if a > 0 and ajb: (9) A positive integer p is called prime if it has just two divisors, namely 1 and p. The unending sequence of primes starts thus: 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37;:::. The Fundamental Theorem of Arithmetic: There is only one way to write a positive integer as a product of primes in nondecreasing order. Examples: 12 = 2 · 2 · 3, 715 = 5 · 11 · 13. 1 (10) Notation: Write m ? n if gcd(m; n) = 1. This is read as \m is prime to n," and we say m and n are \relatively prime." Of course, if m ? n, then n ? m too. (11) The Euler phi function (or totient function) is defined for positive integers n by φ(n) := #fkj0 < k ≤ n; gcd(k; n) = 1g; i.e., φ(n) is the number of positive integers less than or equal to n that are prime to n. n 1 2 3 4 5 6 7 8 9 10 11 12 φ(n) 1 1 2 2 4 2 6 4 6 4 10 4 If p and q are distinct primes, then φ(p) = p − 1 and φ(pq) = (p − 1)(q − 1). (12) Euler's Theorem: If a ? n, then aφ(n) ≡ 1 (mod n). The special case when n = p, a prime, is known as Fermat's Little Theorem: If a ? p, then ap−1 ≡ 1 (mod p). (13) The Chinese Remainder Theorem (Sun Ts˘u,c. A.D. 350): If m ? n, then a ≡ b (mod mn) () a ≡ b (mod m) and a ≡ b (mod n). (14) The extended Euclidean algorithm finds the greatest common divisor g of positive integers a and b (with a > b) and integers s and t such that g = sa + tb. The calculation may be arranged in a tableau. At the start one has q r s t a 1 0 q b 0 1 where q = ba=bc, the integer quotient of a divided by b. Subsequently row j is calculated as follows. rj = rj−2 − qj−1 · rj−1, sj = sj−2 − qj−1 · sj−1, tj = tj−2 − qj−1 · tj−1, qj = brj−1=rjc. The process ends when the remainder rj = 0. For example, gcd(7469; 2387) = 77 = 8 × 7469 − 25 × 2387: q r s t 7469 1 0 3 2387 0 1 7 308 1 −3 1 231 −7 22 3 77 8 −25 0 The extended Euclidean algorithm may be used to solve ax ≡ b (mod n) when g := gcd(a; n)jb: Calculate g = sa + tn. Then x ≡ sb=g + (n=g)k (mod n) with k = 0; ±1; ±2;:::. When b = 1, this procedure finds reciprocals mod n..