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Unformatted text preview: Introduction to Number Theory 1 c circlecopyrt Eli Biham  August 18, 2010 238 Introduction to Number Theory 1 (10) Division Definition : Let a and b be integers. We say that a divides b , or a  b if ∃ d s.t. b = ad . If b negationslash = 0 then  a  ≤  b  . Division Theorem : For any integer a and any positive integer n , there are unique integers q and r such that 0 ≤ r < n and a = qn + r . The value r = a mod n is called the remainder or the residue of the division. Theorem : If m  a and m  b then m  αa + βb for any integers α, β . Proof : a = rm ; b = sm for some r, s . Therefore, αa + βb = αrm + βsm = m ( αr + βs ), i.e., m divides this number. QED c circlecopyrt Eli Biham  August 18, 2010 239 Introduction to Number Theory 1 (10) Division (cont.) If n  ( a − b ), i.e., a and b have the same residues modulo n : ( a mod n ) = ( b mod n ), we write a ≡ b (mod n ) and say that a is congruent to b modulo n . The integers can be divided into n equivalence classes according to their residue modulo n : [ a ] n = { a + kn : k ∈ Z } Z n = { [ a ] n : 0 ≤ a ≤ n − 1 } or briefly Z n = { , 1 , . . . , n − 1 } c circlecopyrt Eli Biham  August 18, 2010 240 Introduction to Number Theory 1 (10) Greatest Common Divisor Let a and b be integers. 1. gcd( a,b ) (the greatest common divisor of a and b ) is gcd( a, b ) Δ = max( d : d  a and d  b ) (for a negationslash = 0 or b negationslash = 0). Note: This definition satisfies gcd(0 , 1) = 1. 2. lcm( a,b ) (the least common multiplier of a and b ) is lcm( a, b ) Δ = min( d > 0 : a  d and b  d ) (for a negationslash = 0 and b negationslash = 0). 3. a and b are coprimes (or relatively prime ) iff gcd( a, b ) = 1. c circlecopyrt Eli Biham  August 18, 2010 241 Introduction to Number Theory 1 (10) Greatest Common Divisor (cont.) Theorem : Let a, b be integers, not both zero, and let d be the smallest positive element of S = { ax + by : x, y ∈ Z } . Then, gcd( a, b ) = d . Proof : S contains a positive integer because  a  ∈ S . By definition, there exist x, y such that d = ax + by . d ≤  a  , thus there exist q, r such that a = qd + r, ≤ r < d. Thus, r = a − qd = a − q ( ax + by ) = a (1 − qx ) + b ( − qy ) ∈ S. r < d implies r = 0, thus d  a . By the same arguments we get d  b . d  a and d  b , thus d ≤ gcd( a, b ). On the other hand gcd( a, b )  a and gcd( a, b )  b , and thus gcd( a, b ) divides any linear combination of a, b , i.e., gcd( a, b ) divides all elements in S , including d , and thus gcd( a, b ) ≤ d . We conclude that d = gcd( a, b ). QED c circlecopyrt Eli Biham  August 18, 2010 242 Introduction to Number Theory 1 (10) Greatest Common Divisor (cont.) Corollary : For any a, b , and d , if d  a and d  b then d  gcd( a, b )....
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This note was uploaded on 04/14/2011 for the course CS 236506 taught by Professor Yanivcarmeli during the Spring '11 term at Technion.
 Spring '11
 YanivCarmeli

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