TOPIC - NUMBERS -2

 TOPIC - NUMBERS -2
• What are Prime Factorization and Prime factors ?   



• Prime factor
The factors which are prime numbers are called prime factors
• Prime factorization
Prime factorization of a number is the expression of the number as the product of its prime factors
Example 1:
Prime factorization of 280 can be written as 280 = 2 × 2 × 2 × 5 × 7 = 2³ × 5 × 7 and the prime factors of 280 are 2, 5 and 7
Example 2:
Prime factorization of 72 can be written as 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² and the prime factors of 72 are 2 and 3


• How to find out prime factorization and prime factors of a number
Repeated Division Method : In order to find out the prime factorization of a number, repeatedly divide the number by the smallest prime number possible(2,3,5,7,11, ...) until the quotient is 1.
Example 1: Find out Prime factorization of 280

        2

280
2

140
2

 70
5

 35
7

  7
   1

Hence, prime factorization of 280 can be written as 280 = 2 × 2 × 2 × 5 × 7 = 2³ × 5 × 7 and the prime factors of 280 are 2, 5 and 7


Example 2: Find out Prime factorization of 72

        2

72
2

36
2

18
3

 9
3

 3
  1

Hence, prime factorization of 72 can be written as 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² and the prime factors of 72 are 2 and 3
• Important Properties
Every whole number greater than 1 can be uniquely expressed as the product of its prime factors. For example, 700 = 2² × 5² × 7



• Multiples   



Multiples of a whole number are the products of that number with 1, 2, 3, 4, and so on
Example : Multiples of 3 are 3, 6, 9, 12, 15, ...
If a number x divides another number y exactly with a remainder of 0, we can say that x is a factor of y and y is a multiple of x
For instance, 4 divides 36 exactly with a remainder of 0. Hence 4 is a factor of 36 and 36 is a multiple of 4



• What is Least Common Multiple (LCM) and how to find LCM    
• Least Common Multiple (LCM)
Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers
Example: LCM of 3 and 4 = 12 because 12 is the smallest multiple which is common to 3 and 4 (In other words, 12 is the smallest number which is divisible by both 3 and 4)
We can find out LCM using prime factorization method or division method
• How to find out LCM using prime factorization method
Step1 : Express each number as a product of prime factors.
Step2 : LCM = The product of highest powers of all prime factors
Example 1 : Find out LCM of 8 and 14
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization andhow to find out Prime Factorization)
8 = 2³
14 = 2 × 7
Step2 : LCM = The product of highest powers of all prime factors
Here the prime factors are 2 and 7
The highest power of 2 here = 2³
The highest power of 7 here = 7
Hence LCM = 2³ × 7 = 56
Example 2 : Find out LCM of 18, 24, 9, 36 and 90
Step1 : Express each number as a product of prime factors (Reference: Prime Factorization and how to find out Prime Factorization).
18 = 2 × 3²
24 = 2³ × 3
9 = 3²
36 = 2³ × 3²
90 = 2 × 5 × 3²
Step2 : LCM = The product of highest powers of all prime factors
Here the prime factors are 2, 3 and 5
The highest power of 2 here = 2³
The highest power of 3 here = 3²
The highest power of 5 here = 5
Hence LCM = 2³ × 3² × 5 = 360
• How to find out LCM using Division Method (shortcut)
Step 1 : Write the given numbers in a horizontal line separated by commas.
Step 2 : Divide the given numbers by the smallest prime number which can exactly divide at least two of the given numbers.
Step 3 : Write the quotients and undivided numbers in a line below the first.
Step 4 : Repeat the process until we reach a stage where no prime factor is common to any two numbers in the row.
Step 5 : LCM = The product of all the divisors and the numbers in the last line.
Example 1 : Find out LCM of 8 and 14

        2 

8, 14
  4,  7

Hence Least common multiple (L.C.M) of 8 and 14 = 2 × 4 × 7 = 56
Example 2 : Find out LCM of 18, 24, 9, 36 and 90

        2 

18, 24,  9, 36, 90
2 

 9, 12,  9, 18, 45
3 

 9,  6,  9,  9, 45
3 

 3,  2,  3,  3, 15
    1,  2,  1,  1,  5
 

Hence Least common multiple (L.C.M) of 18, 24, 9, 36 and 90 = 2 × 2 × 3 × 3 × 2 × 5 = 360



• What is Highest Common Factor (HCF) or Greatest Common Measure (GCM) or Greatest Common Divisor (GCD) and How to find it out ?    



• Highest Common Factor(H.C.F) or Greatest Common Measure(G.C.M) or Greatest Common Divisor (G.C.D)
Highest Common Factor(H.C.F) or Greatest Common Measure(G.C.M) or Greatest Common Divisor (G.C.D) of two or more numbers is the greatest number which divides each of them exactly.
Example : HCF or GCM or GCD of 60 and 75 = 15 because 15 is the highest number which divides both 60 and 75 exactly.
We can find out HCF using prime factorization method or division method
• How to find out HCF using prime factorization method
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization and how to find out Prime Factorization)
Step2 : HCF is the product of all common prime factors using the least power of each common prime factor.
Example 1 : Find out HCF of 60 and 75 (Reference: Prime Factorization and how to find out Prime Factorization)
Step1 : Express each number as a product of prime factors.
60 = 2² × 3 × 5
75 = 3 × 5²
Step2 : HCF is the product of all common prime factors using the least power of each common prime factor.
Here, common prime factors are 3 and 5
The least power of 3 here = 3
The least power of 5 here = 5
Hence, HCF = 3 × 5 = 15
Example 2 : Find out HCF of 36, 24 and 12
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization andhow to find out Prime Factorization)
36 = 2² × 3²
24 = 2³ × 3
12 = 2² × 3
Step2 : HCF is the product of all common prime factors using the least power of each common prime factor.
Here 2 and 3 are common prime factors.
The least power of 2 here = 2²
The least power of 3 here = 3
Hence, HCF = 2² × 3 = 12
Example 3 : Find out HCF of 36, 27 and 80
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization andhow to find out Prime Factorization)
36 = 2² × 3²
27 = 3³
80 = 2⁴ × 5
Step2 : HCF = HCF is the product of all common prime factors using the least power of each common prime factor.
Here you can see that there are no common prime factors.
Hence, HCF = 1
• How to find out HCF using prime factorization method - By dividing the numbers (shortcut)
Step 1 : Write the given numbers in a horizontal line separated by commas.
Step 2 : Divide the given numbers by the smallest prime number which can exactly divide all of the given numbers.
Step 3 : Write the quotients in a line below the first.
Step 4 : Repeat the process until we reach a stage where no common prime factor exists for all of the numbers.
Step 5 :We can see that the factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. Their product is the HCF
Example 1 : Find out HCF of 60 and 75

        3

60, 75
5

20, 25
   4,  5
        

We can see that the prime factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. no common prime factor is exists for the numbers came at the bottom.
Hence HCF = 3 × 5 =15.
Example 2 : Find out HCF of 36, 24 and 12

        2

36, 24, 12
2

18, 12,  6
3

 9,  6,  3
   3,  2,  1
        

We can see that the prime factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. no common prime factor is exists for the numbers came at the bottom.
Hence HCF = 2 × 2 × 3 = 12.
Example 3 : Find out HCF of 36, 24 and 48

        2

36, 24, 48
2

18, 12, 24
3

 9,  6, 12
   3,  2,  4
        

We can see that the prime factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. no common prime factor is exists for the numbers came at the bottom.
Hence HCF = 2 × 2 × 3 = 12.
• How to find out HCF using division method (shortcut)


• To find out HCF of two given numbers using division method
Step 1: Divide the larger number by the smaller number
Step 2: Divisor of step 1 is divided by its remainder
Step 3: Divisor of step 2 is divided by its remainder. Continue this process till we get 0 as remainder.
Step 4: Divisior of the last step is the HCF.


• To find out HCF of three given numbers using division method
Step 1: Find out HCF of any two numbers. 
Step 2: Find out the HCF of the third number and the HCF obtained in step 1
Step 3: HCF obtained in step 2 will be the HCF of the three numbers


• To find out HCF of more than three numbers using division method
In a similar way as explained for three numbers, we can find out HCF of more than three numbers also
Example 1 : Find out HCF of 60 and 75

	60)  75  (1
      	      

60
15)  60  (4
                    

60
0

Hence HCF of 60 and 75 = 15
Example 2 : Find out HCF of 12 and 48

	12)  48  (4
      	      

48
0

Hence HCF of 12 and 48 = 12
Example 3 : Find out HCF of 3556 and 3224

	3224)  3556  (1
      	          

3224
332)  3224  (9
                           

2988
236)  332  (1
                                     

236
96)  236  (2
                                            

192
44)  96  (2
                                                   

88
8)  44  (5
                                                         

40
4)  8  (2
                                                               

8
0

Hence HCF of 3556 and 3224 = 4
Example 3 : Find out HCF of 9, 27, and 48
Taken any two numbers and find out their HCF first. Say, let's find out HCF of 9 and 27 initially.

	9)  27  (3
      	     

27
0

Hence HCF of 9 and 27 = 9
HCF of 9 ,27, 48 = HCF of [(HCF of 9, 27) and 48] = HCF of [9 and 48]

	9)  48  (5
      	     

45
3)  9  (3
                   

9
0

Hence, HCF of 9 ,27, 48 = 3
Example 4 : Find out HCF of 5 and 7

	5)  7  (1
      	     

5
2)  5  (2
                  

4
1)  2  (2
                       

2
0

Hence HCF of 5 and 7 = 1

• How to calculate LCM and HCF for fractions 



Least Common Multiple (L.C.M.) for fractions
LCM for fractions = LCM of NumeratorsHCF of Denominators
Example 1: Find out LCM of 12, 38, 34
LCM = LCM (1, 3, 3)HCF (2, 8, 4)=32



Example 2: Find out LCM of 25, 310
LCM = LCM (2, 3)HCF (5, 10)=65


Highest Common Multiple (H.C.F) for fractions


HCF for fractions = HCF of NumeratorsLCM of Denominators


Example 1: Find out HCF of 35, 611, 920
HCF = HCF (3, 6, 9)LCM (5, 11, 20)=3220


Example 2: Find out HCF of 45, 23
 HCF = HCF (4, 2)LCM (5, 3)=215
 HCF = HCF (4, 2)LCM (5, 3)=215

• How to calculate LCM and HCF for Decimals 



Step 1 : Make the same number of decimal places in all the given numbers by suffixing zero(s) in required numbers as needed.
Step 2 : Now find the LCM/HCF of these numbers without decimal.
Step 3 : Put the decimal point in the result obtained in step 2 leaving as many digits on its right as there are in each of the numbers.
Example1 : Find the LCM and HCF of .63, 1.05, 2.1
Step 1 : Make the same number of decimal places in all the given numbers by suffixing zero(s) in required numbers as needed.
i.e., the numbers can be writtten as .63, 1.05, 2.10
Step 2 : Now find the LCM/HCF of these numbers without decimal.
Without decimal, the numbers can be written as 63, 105 and 210 .
LCM (63, 105 and 210) = 630
HCF (63, 105 and 210) = 21
Step 3 : Put the decimal point in the result obtained in step 2 leaving as many digits on its right as there are in each of the numbers.
i.e., here, we need to put decimal point in the result obtained in step 2 leaving two digits on its right.
i.e., the LCM (.63, 1.05, 2.1) = 6.30
HCF (.63, 1.05, 2.1) = .21



• How to compare fractions? 

  • Type 1 : Fractions with same denominators.
  Compare 35 and 15
  These fractions have same denominator. So just compare the numerators. Bigger the numerator, bigger the number.
  3 > 1. Hence 35>15
  Example 2: Compare 27 and 37 and 87
  These fractions have same denominator. So just compare the numerators. Bigger the numerator, bigger the number.
  8 > 3 > 2. Hence 87>37>27
  
  
  • Type 2 : Fractions with same numerators.
  Example 1: Compare 35 and 38
  These fractions have same numerator. So just compare the denominators. Bigger the denominator, smaller the number.
  8 > 5. Hence 38<35
  Example 2: Compare 78 and 72 and 75
  These fractions have same numerator. So just compare the denominators. Bigger the denominator, smaller the number.
  8 > 5 > 2. Hence 78<75<72
  
  
  • Type 3 : Fractions with different numerators and denominators.
  Example 1: Compare 35 and 47
  To compare such fractions, find out LCM of the denominators. Here, LCM(5, 7) = 35
  Now , convert each of the given fractions into an equivalent fraction with 35 (LCM) as the denominator.
  The denominator of 35 is 5.    5 needs to be multiplied with 7 to get 35. Hence,
  35=3×75×7=2135
  The denominator of 47 is 7.    7 needs to be multiplied with 5 to get 35. Hence,
  47=4×57×5=2035
  2135>2035
  Hence, 35>47
        Or
  Convert the fractions to decimals
  35=.6
  47=.5... (Need not find out the complete decimal value; just find out up to what is required for comparison. In this case the first digit itself is sufficient to do the comparison)
  .6 > .5...
  Hence, 35>47

• Co-prime Numbers or Relatively Prime Numbers    

Two numbers are said to be co-prime (also spelled coprime) or relatively prime if they do not have a common factor other than 1. i.e., if their HCF is 1.
Example1: 3, 5 are co-prime numbers (Because HCF of 3 and 5 = 1)
Example2: 14, 15 are co-prime numbers (Because HCF of 14 and 15 = 1)
A set of numbers is said to be pairwise co-prime (or pairwise relatively prime) if every two distinct numbers in the set are co-prime
Example1 : The numbers 10, 7, 33, 13 are pairwise co-prime, because HCF of any pair of the numbers in this is 1.
HCF (10, 7) = HCF (10, 33) = HCF (10, 13) = HCF (7, 33) = HCF (7, 13) = HCF (33, 13) = 1.
Example2 : The numbers 10, 7, 33, 14 are not pairwise co-prime because HCF(10, 14) = 2 ≠ 1 and HCF(7, 14) = 7 ≠ 1.
If a number is divisible by two co-prime numbers, then the number is divisible by their product also.

Example

3, 5 are co-prime numbers (Because HCF of 3 and 5 = 1)

14325 is divisible by 3 and 5.
3 × 5 = 15
Hence 14325 is divisible by 15 also


If a number is divisible by more than two pairwise co-prime numbers, then the number is divisible by their product also.

Example1 : The numbers 3, 4, 5 are pairwise co-prime because HCF of any pair of numbers in this is 1

1440 is divisible by 3, 4 and 5.

3 × 4 × 5 = 60. Hence 1440 is also divisible by 60

Example2

The numbers 3, 4, 9 are not pairwise co-prime because HCF (3, 9 ) = 3 ≠ 1

1440 is divisible by 3, 4 and 9.

3 X 4 X 9 = 108. However 1440 is not divisible by 108 as 3, 4, 9 are not pairwise co-prime



• Important Points to Note on LCM and HCF   



Product of two numbers = Product of their HCF and LCM.

Example

LCM (8, 14) = 56

HCF (8, 14) = 2

LCM (8, 14) × HCF (8, 14) = 56 × 2 = 112

8 × 14 = 112

Hence LCM (8, 14) × HCF (8, 14) = 8 × 14