Chapter Number -03

PLAYING WITH NUMBERS

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WHEREVER THERE IS NUMBER, THERE IS A BEAUTY.

TOPICS TO BE COVERED

• FACTORS
• MULTIPLES
• PRIME AND COMPSITE NUMBERS
• EVEN AND ODD NUMBERS
• RULES OF Divisibility for 2,3,4,5,6,7,8,9,10 & 11
• Common factors and common multiples
• Prime factorization
• Hcf (highest common factor)
• Lowest common multiple(lcm)
• Problems on hcf and lcm
• What have we discussed?
• Self assessment test
• Pisa (critical & creative thinking test)
• activities

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FACTORS

• A Factor of a number is an exact divisor of that number.

Example:

• 1 x 12 =12, SO 1 and 12 are factor of 12
• 2X6=12, SO 2 and 6 are factor of 12.
• 3X4 =12,SO 3 and 4 are factor of 12
• Therefore , The FACTORS OF 12 are 1,2,3,4,6 and 12

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We know that

• i) 1 is factor of every number.
• Ii) Every number is a factor of itself.
• iii) Every factor is less than equal to the given number.
• iv) Number of factors of a given are finite.

MULTIPLES

• A multiple is the result of multiplying a number by an integer.
• EXAMPLE :
• 12 is a multiple of3, because 3X4=12.
• 12 is also a multiple of 2, because 2X6=12.
• Therefore , the multiples of 12 are 12,24,36,………..

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• PROPERTIES:
• I) The number of multiples of given number is infinite.
• Ii) Every number is a multiple of itself.
• iii) Every multiple of a number is greater than or equal to that number.

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PERFECT NUMBER

• A number for which sum of all its factors is equal to twice the number is called a Perfect number.
• Example: 6 and 28
• The factors of 6 are 1,2,3,6
• Sum of all factors of 6 is 1+2+3+6=12 = 2 X 6
• Factors of 28 are 1,2,4,7,14,28
• Sum of all factors of 28 :

1+2+4+7+14+28 = 56 = 2 X 28

PRIME AND COMPOSITE NUMBERS

• Numbers having exactly two factors(1 and itself) are called PRIME NUMBERS.
• Examples:2,3,5……….. These are having only two factors
• 2 factors are 1 and 2, 3 factors are 1 and 3 factors of 5=1 and 5
• Numbers having more than two factors are called COMPOSITE NUMBERS
• Examples:4,6,8,………………..are having only more than two factors

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IMPORTANT POINTS:

• 1 is neither a prime nor a composite number.
• 2 is the smallest prime number which is even.
• every prime number except 2 is odd.
• 4 is the smallest composite number.
• Two numbers having only 1 as a common factor are called co-prime numbers. So, 4 and 15 are CO-PRIME NUMBERS.
• EX: (2,3) (3,5)…………………
• If the difference between two primes is two then they are called TWIN PRIME NUMBERS
• Ex : (3,5) (5,7) ,,,,,,,,,,

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Sieve of Eratosthenes (Finding for prime numbers)�

• STEP 1 : Cross out 1 because it is not a prime number.
• Step 2 : Encircle 2, cross out all the multiples of 2, other than 2 itself, i.e. 4, 6, 8 and so on.
• Step 3 : You will find that the next uncrossed number is 3. Encircle 3 and cross out all the multiples of 3, other than 3 itself.
• Step 4 : The next uncrossed number is 5. Encircle 5 and cross out all the multiples of 5 other than 5 itself.
• Step 5 : Continue this process till all the numbers in the list are either encircled or crossed out. All the encircled numbers are prime numbers. All the crossed out numbers, other than 1 are composite numbers.
• This method is called the Sieve of Eratosthenes.
• The primes are :2,3,5,7,11,13,17,19,23,29
• 31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
• Total:25

EVEN AND ODD NUMBER

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• Pattern 1 : 2,4,6,8,10,12,14…………… ( Even numbers )
• Pattern 2 : 1,3,5,7,9,11…………… …… ( Odd numbers )

EVEN NUMBER

• The numbers which are exactly divisible by 2(remainder 0) are called an even numbers.
• Example: 2,4,6,8………………….

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ODD NUMBER

• The numbers which are not multiples of 2 are called an odd numbers.
• Example:1,3,5,7,9,…………….
• Note:
• i) The smallest even number is 2.The largest even number is indefinite
• ii)The smallest odd number is 1. The largest odd number is indefinite

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RULES OF DIVISIBILITY

Objective:

• To use rules of divisibility for larger numbers.
• A divisibility rule is a shorthand way of determining whether a given number is divisible by affixed divisor without performing the division, usually by examining its digits.
• Martin Gardner ( American popular Mathematics and science writer) explained and popularized these rules in a book” Mathematical Games”
• Definition:
• Divisibility: The ability to determine what numbers a larger number is divisible by evenly.
• There are some tricks that we can use to help us determine what numbers larger numbers are divisible by without doing long division.
• Let us go for more discussion

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RULE : DIVISIBILITY by 2

• A number is divisible by 2, if the ones or units place is an even that is a number with 0,2,4,6,8, at ones place .

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• Therefore ,any number that ends with 0,2,4,6 or 8 is divisible by 2

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• Example : 78,102,1046………
• These numbers are divisible by 2 because ones places are 8,2 and 6 respectively.

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• Suppose take 2359, Is it divisible by 2 ?
• No, because its unit place is 9 not disable by 2.

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RULE: DIVISIBILITY by 3

• A number is divisible by 3,if the sum of the digits is a multiple of 3.
• Example:
• Is 96 Divisible by 3 ?
• In order to find out , we must add the digits of number 96

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• 9+6= 15, Is 15 multiple of 3 ??? Yes , which means that 96
• Is divisible by 3.

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RULES OF DIVISIBILITY FOR 3 �COMTINUED……

• How about 108 …. Is the number divisible by 3???

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• ADD THE DIGITS TOGETHER
• 1+0+8= 9 is the multiple of 3, so it is divisible by 3.

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• Ex: 199
• Sum of digits 1+9+9=19 is not divisible by 3.
• Therefore,199 is not divisible by 3.

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RULE: DIVISIBILITY by 4

• A Number is divisible by 4,if the last two digits ( ones and tens places ) are divisible by 4 .

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Let's take : 124 for EXAMPLE:

Check 124 is divisible by 4 ? The last two digits in 124 are 24

IS 24 Divisible by 4 ?? YES , This means that 124 is divisible by 4.

• How about 1,312 is the number divisible by 4???

The Last Two Digits in 1,312 are 12

Is 12 divisible by 4????

Yes , Again ,This means that 1,312 is divisible by 4.

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RULE: DIVISIBILITY by 5

• A Number is divisible by 5, If the units place is either 0 (or) 5.

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• So, Any Number that ends with a 0 or a 5 is divisible by 5

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• EXAMPLES : 10.45.95.115.560……

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• These numbers are divisible by 5 because the numbers end with either 0 or 5

RULE: DIVISIBILITY by 6

• If a number is divisible by 2 and 3 both then that number is divisible by6.
• Example:
• number-18
• This number is divisible by 2 and 3 and also both
• Therefore, 18 is divisible by 6.
• Other Example: 27
• This number is only divisible by 3 but not 2. So, 27 is not divisible by 6.

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RULE: DIVISIBILITY by 7

• A number is divisible by 7, we have to understand the following 4 steps:

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• I) Take the last digit of a given number
• II) Double the last digit
• III) Then subtract it from the rest of the number
• Iv) If the resulting number is evenly divisible by 7
• Therefore, the given number is divisible by 7.

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• Example : 203
• (i) The last digit of 203 is 3
• (ii) Double 3 ,means 3 x2=6
• (iii) Subtract 6 from 20 20-6 =14
• (iv) The resulting number is 14, this number is divisible by 7
• Therefore, the given number 203 is divisible by 7.

RULE : DIVISIBILITY by 8

• A number is divisible by 8, if the last three digits are divisible by 8.
• Example(a) 9640

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The last three digits are 640

and we know that, 640÷8=80

so the given number , 9,640 is divisible by 8

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(b). 77184

The last three digits are 184 and this number divisible by 8,184÷8=23

So, 77184 passes this divisibility test.

RULE: DIVISIBILITY by 9

• A number is divisible by 9, if the sum all of the digits is divisible by 9.

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• Example: (a). 702
• add the digits : 7+0+2 = 9
• This number is divisible by 9
• Therefore the given number 702 is divisible by 9.
• Example. (b).5283
• Sum of all digits: 5+2+8+3=18,is divisible by 9
• So, 5283 is divisible by 9.

RULE : DIVISIBILITY by 10

A number is divisible by 10, if the last digit is 0.

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• Example: 7620

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last digit of given number 7620 is 0

so, the number 7620 is divisible by 10

• Example: 32500

The last digit is 0

Therefore, the number 32500 is divisible by 10

RULE : DIVISIBILITY by 11

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• A number is divisible by 11, when the difference of sum of the odd place digits and the sum of even place digits is divisible by 11.
• ( The result must be either zero or multiple of 11)

Example: 5181913

odd place digits : 5+8+9+3=25

Even place digits : 1+1+1=3

Difference: 25-3=22

and 22/11=2 (that is 22 is divisible by 11)

so, 5181913 is divisible by 11.

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COMMON FACTORS AND COMMON MULTIPLES

COMMON FACTORS: The common factors of 4 and 18

• The factors of 4 are : 1, 2 and 4.
• The factors of 18 are : 1, 2, 3, 6, 9 and 18.
• The common factors of 4 and 18 are : 1 and 2
• They are the common factors of 4 and 18.
• Two numbers having only 1 as a common factor are called Co-Prime numbers

Example: 7 and 15 , 12 and 49 , 18 and 23 are co-prime numbers.

• COMMON MULTIPLES: The common multiples of 3, 5 and 6.
• Multiples of 3 are :3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 39,42,45,48,51,54,57,60,.
• Multiples of 5 are :5, 10, 15, 20, 25, 30, 35,40,45,50,55,60, ...
• Multiples of 6 are :6, 12, 18, 24, 30, 36,42,48,54,60,...
• The Common multiples of 3, 5 and 6 are: 30, 60, ...............

PRIME FACTORISATION

• When a number is expressed as product of its prime factors is known as prime factorisation method.
• Finding prime factors ,we have two ways, they are (1) Division method

(2) Tree diagram method

• Example: 24 = 2X 12 = 2 x 2 x 6 = 2 x 2 x 2 x 3 ( 2 and 3 are prime factors)
• 36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3 ( 2 and 3 are prime factors)
• 980 = 2 x 2 x 5 x 7 x 7 (2,5 and 7 are prime factors)

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DIVISION METHOD TREE DIAGRAM METHOD

H C F (HIGHEST COMMON FACTOR)

• We can find the common factors of any two or more numbers, We now try to find the highest of these common factors.
• Example : 12 and 16
• Factors of 12: 1,2,3,4,6,12.
• Factors of 16: 1,2,4,8,16.
• The common factors are 1,2 and 4
• The Highest common factor is 4 , which is called H C F. So HCF (12,16) = 4
• The HCF is the highest number that divides evenly into two or more numbers. It is the largest number possible to simplify those numbers.
• The HCF is also known as G C D ( Greatest Common Divisor)
• Example: 20,28 and 36
• Thus,20 = 2 x 2 x 5 ,28 = 2 x 2 x 7, 36 = 2 x 2 x 3 x 3
• The common factors are: 2 and 2 ( occurring twice )
• The highest common factor is 2 x 2 = 4
• So 4 is the HCF of 20,28 and 36
• HCF(20,28 ,36 ) = 4

H C F : DIVISION METHOD

• To find HCF , we have two methods .They are

(1) Prime factorisation method (2) Division method

• (1) Prime factorisation Method: Just we have discussed this one.
• Resolve into factors, Identify common factors and choose the highest common factor is the HCF.
• (2) Division method:

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LCM: (LOWEST COMMON MULTIPLE)

• We know that, the common multiples of 4 and 6 are 12,24,36,……….. In these multiples, the lowest common multiple is 12.
• Therefore, the lowest common multiple( L C M ) of 4 and 6 is 12.
• The L C M is the lowest common multiple of two or more given numbers is the lowest ( or smallest or least ) of their common multiples,

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LCM : DIVISION METHOD

• Draw a line across and a line down.
• Write the numbers on the line going across to the right.
• Divide the numbers until none can be found.
• Multiply the left column and the bottom row.

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• THUS

LCM=2X2X2X2X3X3X5X2 = 1440

LCM=2X2X2X3X5=120

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What have we discussed?�

• 1. We have discussed multiples, divisors, factors and have seen how to identify factors and multiples.
• 2. We have discussed and discovered the following :
• (a) A factor of a number is an exact divisor of that number.
• (b) Every number is a factor of itself. 1 is a factor of every number.
• (c) Every factor of a number is less than or equal to the given

number.

• (d) Every number is a multiple of each of its factors.
• (e) Every multiple of a given number is greater than or equal to that

number.

• (f) Every number is a multiple of itself.

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• 3. We have learnt that –
• (a) The number other than 1, with only factors namely 1 and the

number itself, is a prime number. Numbers that have more than

two factors are called composite numbers. Number 1 is neither

prime nor composite.

• (b) The number 2 is the smallest prime number and is even. Every

prime number other than 2 is odd.

• (c) Two numbers with only 1 as a common factor are called co-prime

numbers.

• (d) If a number is divisible by another number then it is divisible by

each of the factors of that number.

• (e) A number divisible by two co-prime numbers is divisible by their

product also.

• 4. We have discussed how we can find just by looking at a number, whether it is divisible by small numbers 2,3,4,5,8,9 and 11.
• We have explored the relationship between digits of the numbers and their divisibility by different numbers.
• (a) Divisibility by 2,5 and 10 can be seen by just the last digit.
• (b) Divisibility by 3 and 9 is checked by finding the sum of all digits.

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• (c) Divisibility by 4 and 8 is checked by the last 2 and 3 digits

respectively.

• (d) Divisibility of 11 is checked by comparing the sum of digits at odd

and even places.

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• 5. We have discovered that if two numbers are divisible by a number then their sum and difference are also divisible by that number.

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• 6. We have learnt that –
• (a) The Highest Common Factor (HCF) of two or more given numbers

is the highest of their common factors.

• (b) The Lowest Common Multiple (LCM) of two or more given numbers

is the lowest of their common multiples.

• Relation between given two numbers and its LCM&HCF
• PRODUCT OF TWO NUMBERS = LCM x HCF

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SELF ASSESSMENT TEST:

• 1. Write all the factors of the following numbers :
• (a) 24 (b) 15 (c) 21
• 2. Write first five multiples of : (a) 5 (b) 8 (c) 9
• 3. Write down separately the prime and composite numbers less than 20.
• 4. What is the greatest prime number between 1 and 10?
• 5. Find the common multiples of 3, 4 and 9.
• 6.Find the common factors of (a) 8, 20 (b) 9, 15
• 7. Find the prime factorization of 980.
• 8. Find the HCF of the following numbers :
• (a) 18, 48 (b) 70, 105, 175 (c) 91, 112, 49
• 9. Find the LCM of(a) 20, 25 and 30. (b) 24 and 90.
• 10. I am the smallest number, having four different prime factors.

Can you find me?

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• 11. Using divisibility tests, determine which of the following numbers are divisible by2,3,4,5,6,8,9,10 & 11:
• (a) 572 (b) 726352 (c) 5500 (d) 6000 (e) 12159

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• 12. Which of the following statements are true?
• (a) If a number is divisible by 3, it must be divisible by 9.
• (b) If a number is divisible by 9, it must be divisible by 3.
• (c) A number is divisible by 18, if it is divisible by both 3 and 6.
• (d) If a number is divisible by 9 and 10 both, then it must be divisible by 90. (e) If two numbers are co-primes, at least one of them must be prime.
• (f) All numbers which are divisible by 4 must also be divisible by 8.
• 13. Write the greatest 4-digit number and express it in terms of its prime factors.
• 14. Write the smallest 5-digit number and express it in the form of its prime factors.

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• 15. Fill in the blanks :
• (a) A number which has only two factors is called a ______.
• (b) A number which has more than two factors is called a ______.
• (c) 1 is neither ______ nor ______.
• (d) The smallest prime number is ______.
• (e) The smallest composite number is _____.
• (f) The smallest even number is ______.
• 16. State whether the following statements are True or False:
• (a) The sum of three odd numbers is even.
• (b) The sum of two odd numbers and one even number is even.
• (c) The product of three odd numbers is odd.
• (d) If an even number is divided by 2, the quotient is always odd.
• (e) All prime numbers are odd.
• (f) Prime numbers do not have any factors.
• (g) Sum of two prime numbers is always even.
• (h) 2 is the only even prime number.
• (i) All even numbers are composite numbers.

Activity

• 1.Observe that 2 × 3 + 1 = 7 is a prime number. Here, 1 has been added to a multiple of 2 to get a prime number. Can you find some more numbers of this type?
• 2.Using Sieve of Eratosthes ,find primes from 100 to 200.
• 3.The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.

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• 4.Verify the formula with some examples:

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PRODUCT OF TWO NUMBERS = LCM x HCF

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Pisa (critical and creative Thinking)

• 1. Rena purchases two bags of fertilizer of weights 75 kg and 69 kg. Find the

maximum value of weight which can measure the weight of the fertilizer exact number of

times.

• 2. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and

77 cm respectively. What is the minimum distance each should cover so that all can cover

the distance in complete steps?

• 3. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.
• 4. Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.
• 5. Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.
• 6. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?
• 7. Three tankers contain 403 liters, 434 liters and 465 liters of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.
• 8. Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.
• 9. Find the smallest 4-digit number which is divisible by 18, 24 and 32.

• 10) In a morning walk, three persons step off together. Their steps measure

80 cm, 85 cm and 90 cm respectively. What is the minimum distance each

should walk so that all can cover the same distance in complete steps?

• 11) Two tankers contain 850 liters and 680 liters of kerosene oil respectively.

Find the maximum capacity of a container which can measure the

kerosene oil of both the tankers when used an exact number of times.

• 12) Find all the prime factors of 1729(Hardy-Ramanujan number) and

arrange them in ascending order. Now state the relation, if any; between

two consecutive prime factors.

• 13) Determine if 25110 is divisible by 45.
• 14) A number is divisible by both 5 and 12. By which other number will that

number be always divisible? 15.

• 15. What is the HCF of two consecutive

(a) numbers? (b) even numbers? (c) odd numbers?

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THANK YOU

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BY: DEEKONDA PRABHAKR (TGT MATHS)

JNV NIZAMSAGR

NIZAMBAD , TELANGANA