Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 13 2. NUMBER SYSTEM In general , numbers can be classified in following manner: ❖ Real N umbers (R): Numbers which can be expressed on number line are called Real no. Examples: A ll these numbers represented below are real number s. Number line is geometrical straight line with arbitrarily defined zero (origin). ❖ Imaginary Numbers: Numbers which can ’ t be represented on number line OR we can say All the numbers whose square is negative are called imaginary numbers. For example: √ − 𝟒 is not a real number as its square is a negative number. ✓ Note: √ − 𝟏 = 𝒊 (it is read as iota), hence √ − 𝟒 = √ 𝟒 × √ − 𝟏 = 𝟐𝒊 ✓ Note: √ − 𝟐 ≠ − √ 𝟐 = √ 𝟐 𝒊 (its square is a negative number) ✓ Remember: 𝒊 = √ − 𝟏 , 𝒊 𝟐 = − 𝟏 , 𝒊 𝟑 = − 𝒊 & 𝒊 𝟒 = 𝟏 Hence, 𝒊 + 𝒊 𝟐 + 𝒊 𝟑 + 𝒊 𝟒 = 𝟎 𝑨𝒍𝒔𝒐 , √ − 𝟒 × √ − 𝟗 ≠ √ 𝟑𝟔 ( 𝒊 𝒆 𝟔 ) it is − √ 𝟑𝟔 = − 𝟔 Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 14 ✓ Note: The number 0 is real as well as imaginary number. ❖ Complex Numbers (C/z) : The combined form of real and imaginary numbers is known as complex numbers. It is denoted by : z = x + iy where x is real part and y is imaginary part of z and x , y R. ❖ Real number are of two types: 1. Rational number 2. Irrational num ber Note: This is why sometimes we can say that real number is a collection of rational as well as irrational number. RATIONAL NUMBER (Q) : A real no. that can be written in the form of 𝒑 𝒒 , 𝒒 ≠ 𝟎 & 𝒑 , 𝒒 ∈ 𝑰 Ex: 𝟏𝟎 , − 𝟏𝟓 , 𝟑 𝟐 , 𝟕 , 𝟏 𝟑 , 𝟎 , − 𝟐𝟎 𝟏𝟑 etc SPECIAL TYPES OF RATIONAL NUMBERS Natural Numbers (N): The counting numbers 1, 2, 3, 4...... are called natural numbers i.e. N= {1, 2, 3, 4, .... , } In other words, Set of all non - fractional numbers from 1 to Properties of Natural Numbers: ▪ Addition of natural numbers are closed, associative, and commutative. ▪ Natural Number multiplication is closed, associative, and commutative. ▪ The identity element of a natural number under addition is zero. ▪ The identity element of a natural number under Multiplication is one. Whole numbers(W) : Natural numbers including zero are called whole numbers i.e. W= {0, 1, 2, 3, 4, ..... , } In other words, Set of all non - fractional numbers from 0 to Properties of Whole Numbers: ▪ Whole numbers are closed under addition and multiplication. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 15 ▪ It obeys the commutative and associative property of addition and multipli cation. ▪ Zero is the additive identity element of the whole numbers. ▪ 1 is the multiplicative identity element. ▪ Every natural number is a whole number but vice versa is not true. For example 0. Integers (I/Z): All counting numbers along with their negative including zero is called integers i.e. I or Z = { – ........., – 3, – 2, – 1,0,1,2,3,....... } Note : 0 is neither positive nor negative integer (a) P ositive Integer (I + ) : {1,2,3,................. ......} (b) Negative Integer (I – ) : {................., – 3, – 2, – 1} (c) Non - positive integers : {................., – 3, – 2, – 1,0} (d) Non - negative integers : {0,1,2,3,4,.....................} = W Properties of Integers: ▪ Integers are closed under addition, subtraction, and multiplication. ▪ The commutative property is satisfied for addition and multiplication of integers. ▪ It obeys the associative property of addit ion and multiplication. ▪ It obeys the distributive property for addition and multiplication. ▪ Additi ve identity of integers is 0. ▪ Multiplicative identity of integers is 1. Even Integers: Integers or numbers which are divisible by 2 Example: 𝟎 , ± 𝟐 , ± 𝟒 , ± 𝟔 , ✓ Note: Even numbers are denoted by the expression 2n, where n is any integer. Odd integers: Integers or numbers which are not divisible by 2 Example: ± 𝟏 , ± 𝟑 , ± 𝟓 , ✓ Note: Odd numbers are denoted by the gene ral expression 2 n – 1 where n is any integer. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 16 Prime numbers: A natural number is said to be prime if it has exactly two distinct factors, namely 1and itself. Example : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ............ etc Note: There are 15 p rime numbers between 1 and 50. There are 25 prime numbers between1 to 100. Ty pes of Prime numbers (a) Co - Prime: Two natural numbers (not necessarily prime) are co - prime, if there H.C.F. is one. Example: (1,2), (1,3), (3,4), (3,10), (3,8), (5,6), (7,8) etc. ✓ Note: These numbers are also called as relatively prime numbers. ✓ Note: Two prime numbers (s) are always co - prime but converse need not be true. ✓ Note: Consecutive numbers are always co - prime numbers. (b) Twin Prime: If the difference between two prim e number is two, then the numbers are twin prime pair or prime pair. Example: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73) , (101, 103), (107, 109), (137, 139) etc. ✓ Note: There are only eight twin prim e numbers from 1 to 100. ✓ Note: Every prime pair other than (3,5) is in the form of (6n – 1, 6n+1), where n is any natural number. ✓ Note: The sum of each twin prime except (3,5) is divisible by 12. IDENTIFICATION OF A PR IME NUMBER Certain note about prime number will do the job easily. N1: Prime numbers never ends with 0, 2, 4, 6 & 8 except 2. N2: Prime numbers never ends with 5 except 5. Step 1: Find approximate square root of given number. Step 2: Divide the given number by prime numbers less than approx imate square root of number. If given number is not divisible by any of this prime number then the number is prime otherwise not. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 17 Example: 571, is it a prime or not? Sol. Approximate square root of 571 = 24. Prime number < 24 are 2, 3, 5, 7, 11, 13, 17, 1 9, & 23. But 571 is not divisible by any of these prime numbers so 571 is a prime number Composite Numbers: A number is said to be composite numbers if it has at least three distinct factors. Or in other words we can say No’s which are not prime are comp osite numbers except 1. Example: 4, 6, 8, 9, 10, 12.... Note: (i) ‘1’ is neither prime nor composite number. (ii) ‘2’ is the only even prime number (iii ) T here are 74 composite numbers from 1 to 100. (iv) ‘4’ is the smallest composite numbe r. (v) ‘4’ is s mallest even composite number (vi) ‘9’ is smallest odd composite number SIMPLEST FORM OF A RATIONAL NUMBER A rational number 𝑝 𝑞 is said to be in simplest form, if p and q are integers having no common factor other than 1 ( or p & q co - primes ) Here q 0. They are also called as equivalent rational numbers. Thus, the simplest form of each of 2 4 , 3 6 , 4 8 , 5 10 , 6 12 etc. is 1 2 Sim ilarly, the simplest form of 𝟔 𝟗 is 𝟐 𝟑 and that of 𝟕𝟔 𝟏 𝟑 𝟑 is 𝟒 𝟕 ❖ FRACTION : A fraction is a number which can be written in the form 𝑎 𝑏 , where both a and b are natural numbers and the number 'a' is called numerator and 'b' is ca lled the denominator of the fraction 𝑎 𝑏 , b 0 .hence every fraction is a rational number. For example : 2 5 , 1 3 , 0 5 , 7 15 , are fractions. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 18 DIFFERENT TYPES OF FRACTIO NS Types Definition Examples Proper Numerator < Denominator 2 9 , 3 7 , 12 29 , ... Improper Numerator > Denominator 7 5 , 29 17 , 17 13 ,... Like same denominator 7 12 , 5 12 , 11 12 , ..... Unlike different denominators 2 3 , 4 5 , 11 13 , 7 8 , .... Unit numerator as 1 1 2 , 1 4 , 1 3 , 1 7 , .... Mixed combination of a whole number and a proper fraction. 3 1 2 , 5 1 3 , 8 1 4 ,... Compound N umerator and D enominator themselves are fractions 2 / 3 5 / 7 , 3 / 5 2 / 9 , .... Equivalent Fractions representing the same portion of the whole. 4 6 = 2 3 = 6 9 Decimal denominato r is any of the number 10,100,1000 etc. 8 10 , 11 100 , 17 1000 , V ulgar denominator is a whole number, other than 10,100,1000 etc. 2 7 , 3 8 , 11 17 ✓ Note: Improper fraction can be written in the form of mixed fractions. ❖ DECIMAL: In algebra, a d ecimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point. The dot in a decimal number is called a decimal point. The digits following the decimal point show s a value smaller than one. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 19 ❖ There are two types of decimal numbers, namely: 1. Terminating decimals: Decimals that terminates/ends . In other words, we can say a decimal which has finite number of digits after decimal place. Examples: 𝟎 𝟒𝟓 , 𝟎 𝟐 , 𝟏𝟐 𝟐𝟓 , 𝟐𝟏𝟑 𝟔𝟐𝟓 etc. ✓ Note: Every term inating decimal can easily be converted into p/q forms, hence are always rational numbers. 2. Non - terminating De cimals: Decimals that don’t terminates. It is of two types: a. Non - terminating repeating (Recurring ) decimals: Decimals that repeats itself. it can be of two types: (i) Pure recurring decimal s : A decimal in which all the digits after the decimal point are repeated Ex. : 0 6 , 0 16 , 0 123 are pure recurring decimals. (ii) Mixed recurring decimals: A decimal in which at least one of the digits after th e decimal point is not repeated and then some digit or digits are repeated Ex : 2 1 6 , 0 3 5 , 0 7 85 are mixed recurring decimals. ✓ Note: N on - terminating repeating decimals can be converted into p/q form. Hence, they are always rational numbers. b. Non - ter minating non repeating decimals : Decimals which neither terminates nor repeats itself. Ex: 567 .142857142857..... , 0.10100100010000.... etc. ✓ Note: These types of decimal can never be converted into p/q form. Hence, they are no t rational numbers. Case I: In orde r to convert a pure recurring decimal to 𝒑 𝒒 form W e follow the following steps : Step I: Obtain the repeating decimal and put it equal to x (say). Step II: Write the number in decimal from by r emoving bar from the top of repeating digit an d listing repeating digits at least twice. Ex: x = 0 8 as x = 0.888 Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 20 Step III: Determine the number of digits having bar on their heads. Step IV: If the repeating decimal has 1 place repetition, multiply by 10; a two - place repe tition. multiply by 100; a three - place repetition, multiply by 1000 and so on. Step V: Subtract the number in step II from the number obtained in step IV. Step VI: Divide both sides of the equation by the coefficient of x. St ep VII: Write the rational number in its simplest form. Ex ample: Convert the 23 43 number in the form 𝑝 𝑞 Sol. Let x = 23 43 Then1 x = 23.434343..... ....(i) Here we have only two repeating digit after decimal so we multiply both sides by10 0. Multiplying both sides of (i) by 100, we get 100 x = 2343.4343.... ....(ii) Subtracting (i) from (ii), we get 100x – x = (2343.4343...) – (23 .4343....) 99x = 2320 𝑥 = 2320 99 Alter method: We have, 23 43 = 23 + 0 43 23 43 = 23 + 43 99 [Using the above rule, we have 0 43 = 43 99 ] 23 43 = 23 × 99 + 43 99 = 2277 + 43 99 = 2320 99 Case II: T o convert a mixed recurring decimal to the 𝒑 𝒒 form W e follow the following steps: Step I: Obtain the mixed recurring decimal and write it equal to x (say). Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 21 Step II: Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point. Step III: Multiply both sides of x by 10 n so that only the repeating decimal is on the right side of the decimal point. Step IV: Use the method of converting pure recurring decimal to the form 𝑝 𝑞 and obtain the value of x. Ex ample: Convert 0.2353535... = 0 2 35 in 𝑝 𝑞 form? Sol. Let x = 0 2 35 . Then 10x = 0 2 35 10x = 2 + 0 35 10 𝑥 = 2 + 35 99 [ ∴ 0 35 = 35 99 ] 10 𝑥 = 2 × 99 + 35 99 10 𝑥 = 198 + 35 99 ⇒ 10x = 233 99 ⇒ 𝑥 = 233 990 DETERMINING THE NATURE OF THE DECIMAL EXPANSIONS OF RATIONAL NUMBERS (a) Let x be a rational number whose decimal expansion terminates. Then we can express x in the form 𝑝 𝑞 , where p and q are co - primes, and the prime factoriza tion of q is of the form 2 m × 5 n . where m, n are non - negative integers (W) (b) Let x = 𝑝 𝑞 be a rational number, such that the prime factorization of q is of the form 2 m × 5 n where m, n are non - negative integers. Then, x has a decimal expansi on which terminates. (c) Let 𝑥 = 𝑝 𝑞 be rational number, such that the prime factorization of q is not of the form 2 m × 5 n . where m, n are non - negative integers. Then, x has a decimal expansion which is non - terminating repeating (recur ring decimal) Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 22 ✓ Note: If a rational number 𝑝 𝑞 terminates as q is of the form 2 m × 5 n , then it will terminate up to minimum (m, n) places of decimal. PROPERTIES OF RATIONAL NUMBERS If 𝑎 𝑏 and 𝑐 𝑑 , 𝑒 𝑓 are three ratio nal numbers then: 1. 𝒂 𝒃 + 𝒄 𝒅 = 𝒄 𝒅 + 𝒂 𝒃 [commutative law of ad ditional] 2. ( 𝑎 𝑏 + 𝑐 𝑑 ) + 𝑒 𝑓 = 𝑎 𝑏 + ( 𝑐 𝑑 + 𝑒 𝑓 ) [associative law of additional] 3. ( 𝒂 𝒃 + − 𝒂 𝒃 ) = ( − 𝒂 𝒃 + 𝒂 𝒃 ) = 𝟎 , − 𝒂 𝒃 is called the additive inverse of 𝑎 𝑏 4. ( 𝒂 𝒃 × 𝒄 𝒅 ) = ( 𝒄 𝒅 × 𝒂 𝒃 ) [commutative law of multiplication] 5. ( 𝒂 𝒃 × 𝒄 𝒅 ) × 𝒆 𝒇 = 𝒂 𝒃 ( 𝒄 𝒅 × 𝒆 𝒇 ) [associative law of multiplication] 6. 𝒂 𝒃 × ( 𝒄 𝒅 + 𝒆 𝒇 ) = ( 𝒂 𝒃 × 𝒄 𝒅 ) + ( 𝒂 𝒃 × 𝒆 𝒇 ) [distributive law] TO FIND RATI ONAL NUMBER(S) BETWEEN GIVEN RATIONAL NUMBERS ✓ Note: There are infinitely many rational numbers between any two given rational numbers. Method 1 : Let x and y be two rational numbers such that x < y Then, 1 2 ( 𝑥 + 𝑦 ) is always a rational number between x and y Use above formula repeatedly to get desired rational numbers. Method 2 : Let x and y be two rational numbers such that x < y Suppose we want to find n rational numbers between x and y Let 𝑑 = 𝑦 − 𝑥 𝑛 + 1 Then, n rational n umbers between x and y are: ( 𝑥 + 𝑑 ) , ( 𝑥 + 2 𝑑 ) , ( 𝑥 + 3 𝑑 ) , , ( 𝑥 + 𝑛𝑑 ) Method 3: Making Denominator same by taking LCM. Method 4: Converting given rational numbers into decimals. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 23 ❖ Irrational Numbers (Q c ) : The numbers whose decimal representation is neithe r terminating nor repeating. Also these numbers can never be written in p/q form. Ex: √ 𝟐 , 𝟑 √ 𝟐 , 𝟐 + √ 𝟑 , , 1.327185 ..., 0.10100100010000.... etc. DIFFERENCE BETWEEN RATIONAL AND IRRATIONAL NUMBERS Rational N umber (Q) Irrational N um ber (Q c ) • They can be written in 𝑃 𝑄 form where Q 0 and P, Q ∈ I Ex - 1, 3, 0, − 1 5 , 1 3 , 1 25 etc. • Even No, Odd no, Integers, Natural No, Whole Numbers, P rime No, Composite No, Fractions are all examples of rational no. • √ 𝐶𝑜𝑚𝑝 𝑁𝑜 = 𝑃𝑒𝑟 𝑆𝑞𝑢𝑎𝑟𝑒 is always rational number Ex - √ 4 , √ 9 , √ 16 , , etc. • √ 𝑎 𝑛 if not a surd then it is rational. Ex: √ 27 3 , √ 16 4 etc. • Terminating and repeating decimal are always rational Ex - 1.21, 1 67 , 1 2 31 , 1.5 etc • These numbers can’t be written in 𝑃 𝑄 form. Ex - √ 2 , √ 3 , √ 5 , e tc. 1.0100100010001... etc. • √ 𝑃𝑟𝑖𝑚𝑒 𝑁𝑢 𝑚 𝑏𝑒𝑟 is always irrational number. Ex - √ 2 , √ 3 , √ 5 , etc. • √ 𝐶𝑜𝑚𝑝 𝑁𝑜 ≠ 𝑃𝑒𝑟 𝑆𝑞𝑢𝑎𝑟𝑒 is always irrational No. Ex. √ 6 , √ 8 , √ 10 , etc. • √ 𝑎 𝑛 if it is a surd then it is always irrational No. Ex: √ 9 3 , √ 16 4 , 2 √ 10 3 , √ 32 4 etc. • N on - Terminating and non - repeating decimals are always irrational Ex - 𝜋 = 3 1415 𝑒 = 2 71 4537.82572481923745.... Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 24 ✓ Note: π is an irrational No., While 𝟐𝟐 𝟕 is a rational No. π is approximately written equal to 𝟐𝟐 𝟕 ✓ Note: Between two real number (Q/Q c ) there exists infinitely many real no(Q/Q c ) on number line. OPERATIONS ON REAL NUMBERS Q for Rational n umbers like 0 ,1, 7,3,4,10 , 𝟒 𝟓 , 𝟐 𝟕 , 𝟑𝟖 𝟑𝟓 etc. Q c for Irrational n umbers like √ 𝟐 , √ 𝟑 , √ 𝟔 , √ 𝟑 𝟑 , 𝟐 √ 𝟑 , 𝟐 + √ 𝟑 , 𝟐 − √ 𝟑 Operations Result Examples 𝑸 + 𝑸 Always Q 𝟒 𝟓 + 𝟐 𝟕 = 𝟑𝟖 𝟑𝟓 , 7 + 3 = 10 𝑸 − 𝑸 Always Q 𝟒 𝟓 − 𝟐 𝟕 = 𝟏𝟖 𝟑𝟓 , 7 – 3 = 4 𝑸 + 𝑸 𝒄 Always Q c 𝟐 + √ 𝟑 , 4 + √ 𝟑 𝟑 𝑸 − 𝑸 𝒄 Always Q c 𝟐 − √ 𝟑 , 4 − √ 𝟑 𝟑 𝑸 𝒄 + 𝑸 𝒄 May be Q or Q c √ 𝟑 + 𝟐 √ 𝟑 = 𝟑 √ 𝟑 , √ 𝟑 + √ 𝟓 ( 𝟐 + √ 𝟑 ) + ( 𝟐 − √ 𝟑 ) = 𝟒 𝑸 𝒄 − 𝑸 𝒄 May be Q or Q c 𝟐 √ 𝟑 − √ 𝟑 = √ 𝟑 , √ 𝟑 − √ 𝟑 = 𝟎 𝑸 × 𝑸 Always Q 𝟒 𝟓 × 𝟐 𝟕 = 𝟖 𝟑𝟓 , 7 × 3 = 21 𝑸 × 𝑸 𝒄 May be Q or Q c 2 × √ 𝟑 = 𝟐 √ 𝟑 , 0 × √ 𝟑 = 𝟎 𝑸 𝒄 × 𝑸 𝒄 May be Q or Q c √ 𝟐 × √ 𝟑 = √ 𝟔 , √ 𝟐 × √ 𝟐 = 𝟐 𝑸 ÷ 𝑸 Q or Not defined 𝟒 𝟓 ÷ 𝟐 𝟕 = 𝟏𝟒 𝟓 , 7 ÷ 0 = ND 𝑸 ÷ 𝑸 𝒄 May be Q or Q c 2 ÷ √ 𝟐 = √ 𝟐 , 0 ÷ √ 𝟐 = 0 𝑸 𝒄 ÷ 𝑸 Q c or Not defined 𝟐 √ 𝟑 ÷ 𝟐 = √ 𝟑 , √ 𝟑 ÷ 𝟎 = ND 𝑸 𝒄 ÷ 𝑸 𝒄 May be Q or Q c √ 𝟔 ÷ √ 𝟑 = √ 𝟐 , √ 𝟐 ÷ √ 𝟐 =1 Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 25 SOME IMPORTANT NOTE N1: Negative of an irrational number is an irrational number. N2 : The product of a non - zero rational number & an irrational number will always be an irrational number. N3: The division of a rati onal number with a non - zero rational number always results into a rational number. N4: The division of an irrational number with a non - zero rational number is always irrational number. N5: Sum, difference , product and division of two irra tional n umbers results into a rational or an irrational number. N6: Every rational as well as irrational number can be represented on number line (this we have studied in our class IX) N7: If a and b are two positive rational numbers such t hat ab is not a perfect square of a rational number, then √ 𝒂𝒃 is an irrational number lying between a and b. CLOSURE PROPERTY OF REAL NUMBERS A real number is closed (under an operation) if and only if the operation on any two real n umbers produces real number. If the operation produces even one number not real , the n that operation is said as not closed. For example: The set of real numbers is closed under addition. If you add two real numbers, you will get another real number. Ther e is no possibility of ever getting anything other than another real number. 5 + 12 = 17, 3½ + 6 = 9½, √ 𝟑 + 𝟐 √ 𝟑 = 𝟑 √ 𝟑 etc. Another example: The set of integers {... - 3, - 2, - 1, 0, 1 , 2, 3 ...} is NOT closed under division. 5 ÷ 2 = 2.5 Since 2.5 is not an integer, closure fails. There are also other examples that fail. Next example: The set of real numb ers is NOT closed under division. 3 ÷ 0 = undefined. Since "undefined" is not a rea l number, closure fails. Division by zero is the ONLY case where closure fails for real numbers. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 26 Note: Some textbooks state that " the real numbers are closed under non - zer o division " which, of course, is true. This statement, however, is not equivalent to the general statement that "the real numbers are closed under division". Always read carefully! CLOSURE PROPERTY TABLE FOR BASIC MATHEMATICAL OPERATIONS Numbers Add (+) Subtract ( - ) Multiply ( × ) Division ( ÷ ) Real Closed Closed Closed Not Closed Rational Closed Closed Closed Not Closed Irrational Not Closed Not Closed Not Closed Not Closed Natural Closed Not Closed Closed Not Closed Whole Closed Not Closed Closed Not Closed Integers Closed Closed Closed Not Closed Fundamental theorem of Arithmetic: Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime f actors occur. In other words, it can be stated as “The prime factorization of a natural number is uniq ue, except for the order of its factors” Theorem: Let P be a prime number. If P divides a 2 , then P divides a , where a is a positive integer. H IGHEST COM MON FACTOR(HCF) Factors : A number is a factor of another, if the former exactly divides the latter without leaving a remainder (remainder is zero) They are also called as divisors. For example: Factors of 10 are 1, 2, 5 & 10. Similarly factors of 24 are 1, 2, 3, 4, 6, 8, 12 & 24. Note: 1 and the number itself is always one of the factors of a g iven number. Note: A prime number has only/exactly two factors. Ex: 2, 3, 5 etc. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 27 Definition: HCF also known as GCM (Greatest common Measure) or GCD (gr eatest common divisor). The HCF of two or more than two positive integers is the highest/gre atest positive integers ( that must be common Factor) that divides each of the given positive integer exactly (remainder is zero) For Example : HCF of 3 and 6 is 3 a s three divides 3 as well as 6. (i) 14 is the largest positive integer that divides 28 and 70 completely; therefore H.C.F. of 28 and 70 is 14. (ii) H.C.F. of 75, 125 and 200 is 25 as 25 divides each of 75, 125 and 200 completely and so on. Note: HCF of two or more prime numbers is equal to the smallest prime numbers in the given prime numbers. Note: The HCF of positive integers is always less than or may be equal to the least/smallest of these integers. METHODS TO FIND HCF I. HCF by Listing Method : The listing method involves the process of listing the factors of the given numbers. For example, find the HCF of 20 and 35. • All possible factors of 36 are 1,2, 3 , 4 , 6,12,18 and 36. • All possible factors of 48 are 1, 2 , 3, 4,6, 8 ,12,1 6 , 24 and 48 The common factors of the given num bers are: 1,2, 3, 4, 6 and 12. The greatest among all other numbers is 12 , so it shall be the HCF of both the numbers. II. By applying the Fundamental theorem of Arithmetic method or Prime Factorization method: When we find HCF by prime factorization method, we are finding the greatest common factor among the prime factors or numbers. Steps to be followed for the method are: Let us use these steps in the example below: For Example: Fi nd the HCF of 36 and 48 ? Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 28 Step 1: Finding prime factors individually: All possible factors of 36 are: 2×2×3×3 All possible factors of 48 are: 2×2×2×2×3×1 Step 2: Choose out the common factors: 2×2×3 Step 3: Multiply all the common factors to get the HCF of the given numbers: Here the given numbers are 36 and 48. The product of the common factors: 2×2×3 = 12. So the HCF for the numbers 36 and 48 is 12. III. Long division method : Follow following steps to find HCF: Step 1: We divide the bigger number by smaller one. Step 2: Divide smaller number in step 1 with remainder obtained in step 1. Step 3: Divide divisor of second step with remainder obtained in step 2. Step 4: We will continue this process till we get remainder z ero and divisor obtained in e nd is the required H.C.F. For Example: let us find HCF 36 ) 48 ( 1 of 36 and 48 36 by this method. 12) 36 (3 HCF 36 IV. Euclid’s Division Algorith m: ×× An algorithm is a series of well - defined steps which gives a procedure for solving a type of problem. Euclid’s Division lemma (EDL) : For any two positive integers a and b , there exist unique integers q and r satisfying : a = bq + r, where 0 r < b also q or r can also be zero. Lemma : A lemma is a proven statement used for proving another statement. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 29 ✓ Note: In the relation a = bq + r, where 0 r < b is nothing but a statement of the long division of number a by number b in which q is the quotient obtained and r is the remainder. Thus, dividend = divisor × quotient + remainder a = bq + r SOME IMPORTANT NOTES ABOUT EDL N1 : EDL helps in proving many divisibi lity properties of integers N2: Euclid’s division lemma and division algorithm are so closely interlinked that people often call former as the division algorithm also. N3: Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero, i.e., b ≠ 0. N4: Euclid’s division algorithm can be used to compute the Highest Common Factor (HCF) of two given positive integers. For Example: Consider positive integers 4 8 and 3 6 Step 1: Taking bigger number (48) as a and smaller number (3 6 ) as b. Express the numbers as: a = bq + r i.e. 48 = 3 6 × 1 + 1 2 Step 2: Now taking the divisor 3 6 and remainder 12 , apply the Euclid’s division lemma to get. 3 6 = 12 × 3 + 0 [Expressing as a = bq + r] We will repeat this step until we get remainder as 0. Step 3 : Since, the remainder = 0 so we can ’ t proceed further. Step 4 : The last divisor is 1 2 and we say H.C.F. of 48 and 3 6 is 1 2 ✓ No te: This method works because If a = bq + r, then in each step H.C.F. of (a, b) = H.C.F. of (q, r) ✓ Note: we can note down that all the above four methods will give same HCF. These methods can also be used to find HCF of more than two Positive integers. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 30 SOME RESULTS OF INTEGERS PROVED BY EDL R1: Every even positive integer is of the form 2 q and that every positive odd integer is of the form 2 q + 1, where q is any integer. R2: S quare of any positiv e integer is either of the form 3m or 3m +1 for some i nteger m. R3: A ny positive odd integer is of the form 6 q + 1, 6 q + 3 or 6 q + 5, where q is some integer. R4: T he cube of any positive integer is either of the form 9 q , 9 q + 1 or 9 q + 8. R5: O nly one of the numbers ( n + 2), n and ( n + 4) is divisible by 3. R6: A ny positive even integer is of the form 4 q , 4 q + 2, where q is some integer. R7: A ny positive odd integer is of the form 4 q + 1 or 4 q + 3, where q is some integer. R8: 𝑛 2 − 𝑛 is divisible by 2 for every +ve integer n R9: M ultiplication of two consecutive positive integers is divisible by 2. R10: O ne and only one out of 𝑛 , 𝑛 + 3 , 𝑛 + 6 , 𝑛 + 9 is divisible by 4 R11: If 𝑎 = 𝑥𝑞 + 𝑎 𝑟 , 𝑏 = 𝑦𝑞 + 𝑏 𝑟 , 𝑐 = 𝑧𝑞 + 𝑐 𝑟 , then t he remainder of 𝑎 × 𝑏 × 𝑐 𝑞 is same as the remainder of 𝑎 𝑟 × 𝑏 𝑟 × 𝑐 𝑟 𝑞 LOWEST COMMON MULTIPLE (LCM) Multiples: The multiple of a number are numbers obtained by multiplying the given number with same or other numbers. For example: Mu ltiples of 6 are 6, 12, 18, 24, 30 etc. or indirectly we can all the numbers which lies in the table of 6. ✓ Note: 0 is a multiple of e very number as the product of 0 multiplied by any number is 0 ✓ Note: Number of multiples of a number is infinite while numb er of its factor is always finite (how bigger may number is). ✓ Note: A number is always a multiple of all its factors. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 31 Definition: LCM of two or more than two positive integers is the lowest/least positive integer (that must be common multiple ) which is di visible by each of the given positive integers completely (means it leaves no remainder). For example: LCM of 3 and 6 is 6. (i) LCM of 12 and 18 is 36 as 36 is the smallest positive integer that is divide d by 12 and 18 completely (ii) LCM of 3 , 6, 9 and 2 4 is 72 a s 72 is divi sible by each of 3 , 6, 9 and 2 4 completely Note: LCM of two or more than two prime numbers is the product of all given prime numbers. Note: The LCM of positive integers is always greater than or may be equal to the greatest/highest of these integers. METHODS TO FIND LCM I. LCM by Listing Method: The listing method involves the process of listing the multiples of the given numbers. For example, To find the LCM of 12 and 18 • All possible multiples of 12 are 1 2, 24, 36, 48 , 60, 72, 84, ... • All possible multiples of 18 are 18, 36, 54, 72, 90, 108, .... The common multiples of the given numbers are: 36, 72,108, ... The lowest among all other numbers is 36 , so it shall be the LCM of 12 and 18. You can check 36 is exactly divisible b y 12 and 18. II. By prime Factor ization method: The steps are same as followed in finding HCF. In HCF we take minimum power of all prime factors involved while in LCM we take maximum powers of all the prime factors involved. For Example: to find LCM of 36 and 48 we prime factorize both numbers. 36 = 2×2×3×3 = 2 2 × 3 2 , Similarly 48 = 2 4 × 3 Therefore: LCM (36, 48) = 2 4 × 3 2 = 16 × 9 = 144 III. Ladder or short cut method: We can also find the L.C.M. of the given numbers by dividing all the numbers at the same time by a number that divides at least two of the given numbers. Pocket Book of Maths By : Nab$ Sir Number System AMU Booster Page No. 32 a. When a number is not exactly divisible, we write the number itself below the line. b. When we cannot divide the numbers by a common factor exactly we discontinue dividing the num ber. L.C.M. = 2 × 2 × 2 × 3 × 2 = 48 NOTE: For any two positive integers: Their L.C.M. × their H.C.F. = Product of the number (i) LCM = 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 𝐇𝐂𝐅 (ii) HCF = 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐨𝐟 𝐧𝐮𝐦𝐛𝐞𝐫𝐬 𝐋𝐂𝐌 (iii) One number = 𝐋𝐂𝐌 × 𝐇𝐂𝐅 𝐎𝐭𝐡𝐞𝐫 𝐍𝐮𝐦𝐛𝐞𝐫 HCF AND LCM OF FRACTIONS HC F of fractions = 𝑯𝑪𝑭𝒐𝒇 𝑵𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓𝒔 𝑳𝑪𝑴 𝒐𝒇 𝑫𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓𝒔 For example: The HCF of 4 3 , 4 9 , 2 15 , 36 21 = 𝐻𝐶𝐹𝑜𝑓 ( 4 , 4 , 2 , 36 ) 𝐿𝐶𝑀𝑜𝑓 ( 3 , 9 , 15 , 21 ) = 2 315 LCM of fractions = 𝑳𝑪𝑴 𝒐𝒇 𝑵𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓𝒔 𝑯𝑪𝑭 𝒐𝒇 𝑫𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓𝒔 For example: The LCM of 4 3 , 4 9 , 2 15 , 36 21 = 𝐿𝐶𝑀 𝑜𝑓 ( 4 , 4 , 2 , 36 ) 𝐻𝐶𝐹 𝑜𝑓 ( 3 , 9 , 15 , 21 ) = 36 3 = 12 Note: For any three positive integers p, q and r: HCF (P, q, r) × LCM (p, q, r) p × q × r