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Theory of Numbers

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Theory of Numbers Unit 1 Theory of Numbers Natural Numbers Thenumbers1,2,3,…,whichareusedincountingarecalled naturalnumbers or positiveintegers. Basic Properties of Natural Numbers In the system of natural numbers, we have two ‘operations’ addition and multiplication with the followingproperties. Let x,, y z denotearbitrarynaturalnumbers,then 1. x+ y isanaturalnumber i.e., thesumoftwonaturalnumberisagainanaturalnumber. 2. Commutative law of addition x+ y = y + x 3. Associative law of addition x+()() y + z = x + y + z 4. x⋅ y isanaturalnumber i.e., productoftwonaturalnumbersisanaturalnumber. 5. Commutative law of multiplication x⋅ y = y ⋅ x 6. Associative law of multiplication x⋅()() y ⋅ z = x ⋅ y ⋅ z 7. Existence of multiplicative identity a⋅1 = a 8. Distributive law x() y+ z = xy + xz Divisibility of Integers An integerx ≠ 0 divides y, if there exists an integer a such that y= ax and thus we write as x| y ( x divides y). If x doesnotdivides y,wewriteas x\| y (x doesnotdivides y ) [Thiscanalsobestatedas y isdivisibleby x or x isadivisorof y or y isamultipleof x]. Properties of Divisibility 1. x | y and y | z ⇒ x | z 2. x | y and x | z ⇒x| ( ky± lz ) forall k, l∈ z. z = setofallintegers. 3. x | y and y | x ⇒ x= ± y 4. x | y,where x>0, y > 0 ⇒x≤ y 5. x | y ⇒ x | yz foranyinteger z. 6. x | y iff nx | ny,where n ≠ 0. 2 IndianNational MathematicsOlympiad Test of Divisibility Divisibility by certain special numbers can be determined without actually carrying out the process of division.Thefollowingtheoremsummarizestheresult: ApositiveintegerNisdivisibleby • 2ifandonlyifthelastdigit(unit'sdigit)iseven. • 4ifandonlyifthenumberformedbylasttwodigitsisdivisibleby4. • 8ifandonlyifthenumberformedbythelastthreedigitsisdivisibleby8. • 3ifandonlyifthesumofallthedigitsisdivisibleby3. • 9ifandonlyifthesumofallthedigitsisdivisibleby9. • 5ifandonlyifthelastdigitiseither0or5. • 25ifandonlyifthenumberformedbythelasttwodigitsisdivisibleby25. • 125ifandonlyifthenumberformedbythelastthreedigitsisdivisibleby125. • 11 if and only if the difference between the sum of digits in the odd places (starting from right) and sumofthedigitsintheevenplaces(startingfromtheright)isamultipleof11. Division Algorithm For any two natural numbers a and b, there exists unique numbers q and r called respectively quotient andremainder, a= bq + r, where 0 ≤r < b. Common Divisor If a number ‘c’ divides any two numbers a and b i.e., if c | a and c | b, then c is known as a common divisor of a and b. Greatest Common Divisor If a number d divides a and b and is divisible by all the common divisors of a and b, then d is known as the greatestcommondivisor (GCD)of a and b orHCFof a and b. TheGCD ofnumbers a and b istheuniquepositiveinteger d withthefollowingtwoproperties. (i) d | a and d | b (ii) If c | a and c | b;then c | d Wewriteitas (,)a b= d Forexample, (,);12 15= 3 (,)7 8= 1 Note 1. (,)(,)a b= b a 2.If a|b;then (,)a b= a Properties of GCD 1. If (,)b c= g and d isacommondivisorof b and c,then d isadivisorof g. 2. Forany m > 0, (mb , mc )= m ( b , c ) b c 1 3. If d |b and d |c and d > 0,then ,(,) = b c d d d b c 4. If (,)b c= g,then , = 1 g g 5. If (,)b c= g,thenthereexiststwointegers x and y suchthat g= xb + yc 6. If (,)a b = 1 and (,)a c = 1,then (,)a bc = 1 7. If a| bc = 1 and (,),a b = 1 then a | c = 1 TheoryofNumbers 3 Forexample a=6,, b = 21 c = 10 6| 21× 10 but (,)6 21= 3 and (,)6 10= 2 and6dividesneither21nor10 LCM of two integers a, b is the smallest positive integer divisible by both a and b and it is denoted by [a,b]. The Euclidean algorithm can be used to find the GCD of two integers as well as representing GCD as in5thproperty.Consider2numbers18,28 28= 1. 8 + 10 ;18= 110. + 8 10= 1 ⋅ 8 + 2 ; 8= 4 ⋅ 2 + 0 (,)18 28= 2 (,).18 28= 2 = 10 − 1 8 =10 −(.) 18 − 110 =210.. − 118 =2(.). 28 − 118 − 118 =228.. − 318 =228.() + − 3 18 Note Therepresentationin5thpropertyisnotunique.Infactwecanrepresent (,)a b as xa+ yb ininfinite numberofwayswhere x, y∈ Z Z = setofallintegers. Inaboveexample,252isLCMof18and28. 252= 9. 28 252= 1418. (,).()18 28= 2 28 + − 3 ⋅ 18 =228.() + 252KK + − 318 − 252 =()()2 + 9KK 28 + − 3 − 14 18 where K isanyinteger. Unit 1iscalledunitinthesetofpositiveintegers. Prime Apositiveinteger P issaidtobeprime,if (i) P > 1 (ii) P has no divisors except 1 and P i.e., A number which has exactly two different factors, itself and one,iscalleda primenumber. Thus, 2, 3, 5, 7, 11, ... are primes. 2 is the only even number which is prime. All other primes beingodd. Buttheconverseisnottrue i.e., everyoddnumberneednotbeprime. Composite Every number (greater than one) which is not prime is called composite number. i.e., a number which has more than two different factors is called composite. For example 18 is a composite number because 2, 3, 6,9aredivisorsof18otherthan1and18. We can also define a composite number as : A natural number n is said to be composite, if there exists integers l and m suchthat n = lm,where 1 <l < n and l< m < n. 4 IndianNational MathematicsOlympiad Remark G Aprimenumber P canbewrittenasaproductonlyinonewaynamely P.1. G A composite number n can also be written as n.1. But composite number can be written in one more way alsoasmentionedabove. G Acompositenumberhasatleastthreefactors. Note 1isneitherprimenorcomposite. Twin Primes Apairofnumbersissaidtobe twinprimes,iftheydifferby2. e.g., 3,5aretwinprimes. Perfect Number Anumber n issaidtobeperfectifthesumofalldivisorsof n (including n)isequalto2n. Forexample 28isaperfectnumberbecausedivisorsof28are1,2,4,7,14,28. Sumofdivisorsof n ()=28 = 1 + 2 + 4 + 7 +14 + 28 =56 = 2n Coprime Integers Twonumbers a and b aresaidtobe coprime,if1isonlycommondivisorsof a and b. i.e., ifGCD of a and b = 1 i.e., if (,)a b = 1 e.g., (4,5) = 1,(8,9) = 1. Theorem1 If a= qb + r , then (,)(,)a b= b r . Proof Let (,)a b= d and (,)b r= e Q (,)a b= d ∴ d | a and d | b ∴ d | aand d| qb ∴ d | (a –qb) i.e., d |r []Qa− qb = r ∴ d iscommondivisorof b and r. ∴ d |e [Qe istheGCD of b and r]…(i) Again Q (b, r) = e e e ∴ e |b and e |r ∴ and bq r ∴ e|bq + r i.e., e|a []Qa= bq + r ∴ e isacommondivisorof a and b. ∴ e|d [Qd istheGCD of a and b]…(ii) From Eqs.(i)and(ii),wehave d= e i.e., (,)(,)a b= b r Remark Theaboveresultcanalsobestatedas: GCD of a and b issameasGCD of b and r,where r isremainderobtainedondividing a by b. Corollary If (,)a b = 1,then (,)(,)b r= a b = 1 i.e., if a is coprime to b,then r is coprime to b. r isremainderobtainedondividing a by b. TheoryofNumbers 5 Theorem 2 If d is the greatest common divisor of a and b, then there exists integers x and y such that d= xa + yb anddistheleastpositivevalueof xa+ yb. Proof CaseI Bysuccessiveapplicationofdivisionalgorithmtonumbers a and b. K Let r1,,, r 2 rn besuccessiveremainders. = + < < Therefore, a bq1 r 10, r 1 b (Dividing a by b) = + < < b r1 q 2 r 20, r 2 r 1 (Dividing b byremainder r1 ) = + < < r1 r 2 q 3 r 3, 0 r 3 r 2 (Dividing r1 byremainder r2 ) M M M = + < < rn−2 r n − 1 qn r n , 0 rn rn − 1 = + < < rn −1 rn qn + 1 r n + 1, 0 rn + 1 rn > > K Since r1 r 2 is a set of decreasing integers, this process must terminate after a finite number of step. i.e., remaindermustbezeroaftersomestage. = ∴ = + = Solet rn + 1 0 rn −1 rn qn + 10 rn qn + 1 ∴ = = rn | rn − 1 (,)rn − 1 rn r n [If a | b,then ()]a, b a = = = = K = = Now, ()()(,)(,)a,, b b r1 r 1 r 2 r 2 r 3 (,)rn − 1 rn rn …(i) i.e., GCDof a and b is rn . = − = + = = − Fromfirstofaboveequations r1 a bq 1 ax 1 by 1,where x11,, y 1 q = − = − Puttingthevalueof r1 a bq 1 in r2 b r 1 q 2 = − = − − = − + r2 b r 1 q 2 b() a bq 1 q 2 b aq2 bq 1 q 2 = − + + = + = − aq2 b1( q 1 q 2 ) ax2 by 2 , where x2 q 2 = + y21 q 1 q 2 = + = + Similarly, r3 ax 3 by 3 andsoon rn ax n by n = + or rn ax by = = where xn x and yn y = = = + i.e., GCD of a and b rn canbeexpressedas (,)a b d ax by [ByEq.(i)] CaseII (,)a b= d (Q d | a and d | b) ∴ d| ( ax+ yb ) forallvaluesof x and y. ∴ ∃ aninteger k suchthat xa+ yb = kd …(ii) Butleastvalueof k is1. Putting k = 1 inEq.(ii),leastvalueof xa+ yb is d. Corollary If a and b are coprime integers i.e., if (,),a b = 1 then there exists integers x and y such that ax+ by = 1 a b Example 1 If (,)a b= d,then , =1. d d Solution Q (,)a b= d ∴ d| a and d| b [BydefinitionofGCD]. ∴ = Thereexistsintegers a1and b1 suchthat a da1 …(i) = b db1 …(ii) Again Q (,)a b= d 6 IndianNational MathematicsOlympiad Thereexistsintegers x and y suchthat ax+ by = d [ByTheorem2] Puttingvaluesof a and b fromEqs.(i)and(ii) + = da1 x db 1 y d. + = or a1 x b 1 y 1 = (,)a1 b 1 1 [Bycorollarytheorem1] a b = Q =a = b or , 1 FromEq.( i ),a1, fromEq. ( ii ),b 1 d d d d Remark = = = = (,)a b d and a a1 d, b b 1 d,then a1 and b1 are coprime i.e., (,)a1 b 1 1 Example 2 Ifa|bc and (,),a b =1 thena|c. Solution Q a|bc ∴ Thereexistsaninteger d suchthat bc= ad …(i) Q (,)a b =1 ∴ Thereexistintegers m and n suchthat am+ bn =1 …(ii) MultiplyingbothsidesofEq.(ii)by c, acm+ bcn = c …(iii) Putting bc= ad fromEq.(i)inEq.(iii) acm+ adn = c a() cm+ dn = c ∴ a|c Note If a|bc and (,)a b =1,then a|c.Thisresultisalsoknownas GaussTheorem.
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