ACT-Math-2-整數(shù)的概念和性質(zhì)

Arithmetic

算術(shù)第二節(jié)整數(shù)的概念和性質(zhì)Ⅰ.TheConceptofIntegers(整數(shù)的概念)自然數(shù)奇數(shù)偶數(shù)質(zhì)數(shù)合數(shù)互質(zhì)數(shù)倍數(shù)和約數(shù)公倍數(shù)公約數(shù)或公因數(shù)完全平方數(shù)商和余數(shù)連續(xù)整數(shù)NaturalNumbersOddNumbersEvenNumbersPrimeNumbersCompositeNumbersMutualPrimeNumbersMultipleandDivisorCommonMultipleCommonDivisororFactorPerfectSquareQuotientsandRemaindersConsecutiveIntegersⅠ.TheConceptofIntegers(整數(shù)的概念)NaturalNumbers(自然數(shù)):Anyofthenumbers0,1,2,3,4,…thatcanbeusedtocountthemembersofasetOddNumbers(奇數(shù))Numbersthatcannotbeevenlydivisibleby2EvenNumbers(偶數(shù))Numbersthatcanbeevenlydivisibleby2Ⅰ.TheConceptofIntegers(整數(shù)的概念)PrimeNumbers(質(zhì)數(shù))Anaturalnumbergreaterthan1thathasnopositivedivisorsotherthan1anditself.,suchas17.CompositeNumbers(合數(shù))AnaturalnumberthatcanbefactorizedintotwoormoreotherpositiveintegersMutualPrimeNumbers(互質(zhì)數(shù))Twonumberswith1onlyastheirgreatestcommondivisorⅠ.TheConceptofIntegers(整數(shù)的概念)MultipleandDivisor(倍數(shù)和約數(shù))Ifonenumbercanbeexactlydividedbyasmallernumber,thenumberisamultipleofasmallernumberandthesmallernumberisadivisorofthisnumber.\CommonMultiple(公倍數(shù))Ifanumberisdivisiblebytwoormoreintegers,thisnumberisacommonmultipleoftheseintegers.theleast/lowestcommonmultiple最小公倍數(shù)CommonDivisororFactor(公約數(shù)或公因數(shù))Iftwoormoreintegersaredivisiblebyanumber,thisnumberisacommondivisoroftheseintegers.thegreatestcommondivisor最大公約數(shù)Ⅰ.TheConceptofIntegers(整數(shù)的概念)PerfectSquare(完全平方數(shù))Ifthesquarerootofanintegerisstillaninteger,theoriginalintegerisaperfectsquarePerfectcube完全立方數(shù)QuotientsandRemainders(商和余數(shù))Theremainderistheinteger"leftover"afterdividingoneintegerbyanothertoproduceanintegerquotient.ConsecutiveIntegers(連續(xù)整數(shù))Ⅱ.ThePropertiesofIntegers(整數(shù)的性質(zhì))1.PositiveandNegative(正負性)正整數(shù)(positiveintegers)，也就是自然數(shù)(naturalnumbers),如1,2,3,…負整數(shù)(negativeintegers),如…,-3,-2,-10既不是整數(shù)也不是負數(shù)。e.g.:Isxpositive?x2–1=0X3+1=0Ⅱ.ThePropertiesofIntegers(整數(shù)的性質(zhì))2.OddandEven(奇偶性)Ifnisaninteger,then

isanevennumberand

isanoddnumber.Odd±odd=?Even±even=?Odd±even=?Even±even±even±…±even=?Odd×odd×odd×…×odd=?Odd×odd×odd×…×odd×even=?Even×even=?Ⅱ.ThePropertiesofIntegers(整數(shù)的性質(zhì))2.OddandEven(奇偶性)任何一個大于2的偶數(shù)都可表示為兩個質(zhì)數(shù)的和。2個連續(xù)的自然數(shù)相乘必然為2的倍數(shù)，3個連續(xù)的自然數(shù)相乘必然為6的倍數(shù)，若3個連續(xù)自然數(shù)的算術(shù)平均值為奇數(shù)，則這三個自然數(shù)的乘積必為8的倍數(shù)。如：(4+5+6)/3=5，則4×5×6=120可被8整除。ACTPractice1.Ifxisaneveninteger,whichofthefollowingisanoddinteger?(A)3x+2(B)7x(C)8x+5(D)x2(E)x32.Ifaandbareintegersanda–b=8,then(a+b)CANNOTbe(A)0(B)lessthan6(C)greaterthan6(C)aneveninteger(E)anoddintegerACTPractice3.Ifnisanintegergreaterthan2,whichofthefollowingCANNOTbeaneveninteger?(A)n2(B)n(n–1)(C)n–1(D)n+1(E)4n+3Ⅱ.ThePropertiesofIntegers(整數(shù)的性質(zhì))3.PrimeandCompositenumbers(質(zhì)數(shù)和合數(shù))Note:數(shù)字1既不是質(zhì)數(shù)，也不是合數(shù)。大于2的質(zhì)數(shù)都是奇數(shù)，數(shù)字2是質(zhì)數(shù)中唯一的偶數(shù)。定理：任何一個大于2的偶數(shù)都可以表示為兩個質(zhì)數(shù)的和。ACTPractice1.Whichofthefollowingcannotbeexpressedasthesumoftwoprimenumbers.(A)21(B)14(C)18(D)28(E)232.Ifthesumoftwoprimenumbersis40,whatisthelargestproductofthetwointegers?(A)391(B)319(C)111(D)153(E)387Ⅱ.ThePropertiesofIntegers(整數(shù)的性質(zhì))4.DivisorandMultiple(約數(shù)和倍數(shù))如果整數(shù)a能被整數(shù)b整除，則a能被b的約數(shù)整除。0為任何一個非零整數(shù)的倍數(shù)，1為任何一個整數(shù)的約數(shù)，任何一個質(zhì)數(shù)有且只有1和它本身兩個約數(shù)。求最大公約數(shù)和最小公倍數(shù)5.TheDivisibilityofIntegers(整數(shù)的整除特性)Herearesomeshortcutstodeterminingdivisibilitybycommonnumbers:IftheintegerhasthisfeatureThenitisdivisiblebyItendsin0,2,4,6or82Thesumofthedigitsisdivisibleby33Thenumberformedbythelast2digitsisdivisibleby44Thenumberendin5or05Thenumbermeetsthetestsfordivisibilityby2and36Thenumberformedbythelast3digitsisdivisibleby88Thesumofthedigitsisdivisibleby99Thedifferencebetweenthesumofodddigitsandthesumofevendigitsisdivisibleby11116.ThePropertiesofConsecutiveIntegers(連續(xù)整數(shù)的性質(zhì))Anytwoconsecutiveintegersmustbeanoddintegerandaneveninteger.Oneintegeramonganythreeconsecutiveintegerscanbedivisiblebythree;thus,theproductofthesethreeconsecutiveintegerscanbedivisibleby3.Thenumberofconsecutiveintegersn,n+1,n+2,…,n+kisk+1,andtheirsumis(n+k/2)×(k+1)7.TheBasicPropertiesofSquare(平方數(shù)的基本性質(zhì))平方數(shù)的個位是0,1,4,5,6,9之一；偶平方數(shù)能被4整除；奇平方數(shù)能被8整除余1，即它可寫為8k+1，k為整數(shù)；在相鄰的兩個自然數(shù)的平方之間不存在其他的完全平方數(shù)；任何兩個相鄰自然數(shù)之積不是完全平方數(shù)；兩個奇數(shù)的平方之和不是完全平方數(shù)。7.自然數(shù)n次冪的尾數(shù)特征尾數(shù)為2的數(shù)的冪的個位數(shù)一定以2,4,8,6循環(huán)；尾數(shù)為3的數(shù)的冪的個位數(shù)一定以3,9,7,1循環(huán)；尾數(shù)為4的數(shù)的冪的個位數(shù)一定以4,6循環(huán)；尾數(shù)為6的數(shù)的冪的個位數(shù)一定以6循環(huán)；尾數(shù)為7的數(shù)的冪的個位數(shù)一定以7,9,3,1循環(huán)；尾數(shù)為8的數(shù)的冪的個位數(shù)一定以8,4,2,6循環(huán)；尾數(shù)為9的數(shù)的冪的個位數(shù)一定以9,1循環(huán)；Practice：1.Ifxandyaredifferentprimenumbers,eachgreaterthan2,whichofthefollowingmustbetrue?x+y≠91x–yisaneveninteger;x/