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Home , Modular arithmetic

Modular Basics

(1) The “floor” function is defined by the formula bxc := ( the greatest 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 ,” abbreviated gcd, of a set of 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 in integer 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 . 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 behave “as expected” for the operations of , , and . 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 of solutions of ax ≡ b (mod m) in the set {1, 2, . . . , m} is 0 if g does not divide b and is g if g|b, 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 a|b, 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: 7|98; −5|100; but 8 does not divide 26. Graham, Knuth, and Patashnik’s divisibility notation: a\b if and only if a > 0 and a|b.

(9) A positive integer p is called prime if it has just two , 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 of primes in nondecreasing . 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) := #{k|0 < k ≤ n, gcd(k, n) = 1}, 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 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 may be used to solve ax ≡ b (mod n) when g := gcd(a, n)|b: 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.