Thursday, December 18, 2014

number system


  • Number Sets and Properties of Numbers   


    • Counting Numbers (Natural numbers) : 1, 2, 3 ...

    • Whole Numbers : 0, 1, 2, 3 ...

    • Integers : -3, -2, -1, 0, 1, 2, 3 ...

    • Rational Numbers
      • Rational numbers can be expressed as ab where a and b are integers and b≠0
      Examples: 112, 42, 0, −811 etc.
      All integers, fractions and terminating or recurring decimals are rational numbers.
      Counting Numbers (Natural numbers),Whole Numbers, Integers, Rational Numbers

    • Irrational Numbers
      • Any number which is not a rational number is an irrational number. In other words, an irrational number is a number which cannot be expressed as ab where a and b are integers.
      For instance, numbers whose decimals do not terminate and do not repeat cannot be written as a fraction and hence they are irrational numbers.
      Example : π, 2√, (3+5√), 43√ (meaning 4×3√), 6√3 etc
      Please note that the value of π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679...
      We cannot π as a simple fraction (The fraction 22/7 = 3.14.... is just an approximate value of π)

    • Real Numbers
      • Real numbers include counting numbers, whole numbers, integers, rational numbers and irrational numbers.

    • Surds
      • Let a be any rational number and n be any positive integer such that a√n is irrational. Then a√n is a surd.
      Example : 3√, 10−−√6, 43√ etc
      Please note that numbers like 9√, 27−−√3 etc are not surds because they are not irrational numbers
      Every surd is an irrational number. But every irrational number is not a surd. (eg : π , e etc are not surds though they are irrational numbers.)

  • Addition, Subtraction and Multiplication Rules for Even and Odd Numbers    


      • Addition Rules for Even and Odd Numbers
    • The sum of any number of even numbers is always even

    • The sum of even number of odd numbers is always even

    • The sum of odd number of odd numbers is always odd

      Subtraction Rules for Even and Odd Numbers
    • The difference of two even numbers is always even

    • The difference of two odd numbers is always even

      Multiplication Rules for Even and Odd Numbers
    • The product of even numbers is always even

      •
    • The product of odd numbers is always odd

      •
    • If there is at least one even number multiplied by any number of odd numbers, the product is always even

  • Divisibility    Minimize Button


    • Divisible By
      • One whole number is divisible by another if the remainder we get after the division is zero.
      Examples
      • 36 is divisible by 4 because 36 ÷ 4 = 9 with a remainder of 0.
      • 36 is divisible by 6 because 36 ÷ 6 = 6 with a remainder of 0.
      • 36 is not divisible by 5 because 36 ÷ 5 = 7 with a remainder of 1.
    • Divisibility Rules
    By using divisibility rules we can easily find out whether a given number is divisible by another number without actually performing the division. This helps to save time especially when working with numbers.
    Minimize Button
    Divisibility RuleDescriptionExamples
    Divisibility by 2A number is divisible by 2 if the last digit is even. i.e., if the last digit is 0 or 2 or 4 or 6 or 8Example1: Check if 64 is divisible by 2.

    The last digit of 64 is 4 (even). Hence 64 is divisible by 2

    Example2: Check if 69 is divisible by 2.

    The last digit of 69 is 9 (not even). Hence 69 is not divisible by 2
    Divisibility by 3
    A number is divisible by 3 if the sum of the digits is divisible by 3
    (Please note that we can apply this rule to the answer again and again if we need)
    		Example1: Check if 387 is divisible by 3.

    3 + 8 + 7 = 18.
    18 is divisible by 3. Hence 387 is divisible by 3

    Example2: Check if 421 is divisible by 3.

    4 + 2 + 1 = 7.
    7 is not divisible by 3. Hence 421 is not divisible by 3
    Divisibility by 4A number is divisible by 4 if the number formed by the last two digits is divisible by 4.Example1: Check if 416 is divisible by 4.

    Number formed by the last two digits = 16.
    16 is divisible by 4. Hence 416 is divisible by 4

    Example2: Check if 481 is divisible by 4.

    Number formed by the last two digits = 81.
    81 is not divisible by 4. Hence 481 is not divisible by 4
    Divisibility by 5A number is divisible by 5 if the last digit is either 0 or 5.Example1: Check if 305 is divisible by 5.
    Last digit is 5. Hence 305 is divisible by 5.

    Example2: Check if 420 is divisible by 5.
    Last digit is 0. Hence 420 is divisible by 5.

    Example3: Check if 312 is divisible by 5.
    Last digit is 2. Hence 312 is not divisible by 5.
    Divisibility by 6A number is divisible by 6 if it is divisible by both 2 and 3.Example1: Check if 546 is divisible by 6.

    546 is divisible by 2. 546 is also divisible by 3. (Check the divisibility rule of 2 and 3 to find out this)

    Hence 546 is divisible by 6

    Example2: Check if 633 is divisible by 6.

    633 is not divisible by 2 though 633 is divisible by 3. (Check thedivisibility rule of 2 and 3 to find out this)

    Hence 633 is not divisible by 6

    Example3: Check if 635 is divisible by 6.

    635 is not divisible by 2. 635 is also not divisible by 3. (Check thedivisibility rule of 2 and 3 to find out this)

    Hence 635 is not divisible by 6

    Example4: Check if 428 is divisible by 6.

    428 is divisible by 2 but 428 is not divisible by 3.(Check the divisibility rule of 2 and 3 to find out this)

    Hence 428 is not divisible by 6
    Divisibility by 7To find out if a number is divisible by 7, double the last digit and subtact it from the number formed by the remaining digits.

    Repeat this process until we get at a smaller number whose divisibility we know.

    If this smaller number is 0 or divisible by 7, the original number is also divisible by 7.    		Example1: Check if 349 is divisible by 7.

    Given number = 349
    34 - (9 × 2) = 34 - 18 = 16
    16 is not divisible by 7. Hence 349 is not divisible by 7

    Example2: Check if 364 is divisible by 7.

    Given number = 364
    36 - (4 × 2) = 36 - 8 = 28
    28 is divisible by 7. Hence 364 is also divisible by 7

    Example3: Check if 3374 is divisible by 7.

    Given number = 3374
    337 - (4 × 2) = 337 - 8 = 329
    32 - (9 × 2) = 32 - 18 = 14
    14 is divisible by 7. Hence 329 is also divisible by 7.
    Hence 3374 is also divisible by 7.
    Divisibility by 8A number is divisible by 8 if the number formed by the last three digits is divisible by 8.Example1: Check if 7624 is divisible by 8.

    The number formed by the last three digits of 7624 = 624.
    624 is divisible by 8. Hence 7624 is also divisible by 8.

    Example2: Check if 129437464 is divisible by 8.

    The number formed by the last three digits of 129437464 = 464.
    464 is divisible by 8. Hence 129437464 is also divisible by 8.

    Example3: Check if 737460 is divisible by 8.

    The number formed by the last three digits of 737460 = 460.
    460 is not divisible by 8. Hence 737460 is also not divisible by 8.
    Divisibility by 9A number is divisible by 9 if the sum of its digits is divisible by 9.

    (Please note that we can apply this rule to the answer again and again if we need)    		Example1: Check if 367821 is divisible by 9.

    3 + 6 + 7 + 8 + 2 + 1 = 27
    27 is divisible by 9. Hence 367821 is also divisible by 9.

    Example2: Check if 47128 is divisible by 9.

    4 + 7 + 1 + 2 + 8 = 22
    22 is not divisible by 9. Hence 47128 is not divisible by 9.

    Example3: Check if 4975291989 is divisible by 9.
    4 + 9+ 7 + 5 + 2 + 9 + 1 + 9 + 8 + 9= 63
    Since 63 is big, we can use the same method to see if it is divisible by 9.
    6 + 3 = 9
    9 is divisible by 9. Hence 63 is also divisible by 9.
    Hence 4975291989 is also divisible by 9.
    Divisibility by 10A number is divisible by 10 if the last digit is 0.Example1: Check if 2570 is divisible by 10.

    Last digit is 0. Hence 2570 is divisible by 10.

    Example2: Check if 5462 is divisible by 10.

    Last digit is not 0. Hence 5462 is not divisible by 10
    Divisibility by 11To find out if a number is divisible by 11, find the sum of the odd numbered digits and the sum of the even numbered digits.

    Now substract the lower number obtained from the bigger number obtained.

    If the number we get is 0 or divisible by 11, the original number is also divisible by 11.    		Example1: Check if 85136 is divisible by 11.

    8 + 1 + 6 = 15
    5 + 3 = 8
    15 - 8 = 7
    7 is not divisible by 11. Hence 85136 is not divisible by 11.

    Example2: Check if 2737152 is divisible by 11.

    2 + 3 + 1 + 2 = 8
    7 + 7 + 5 = 19
    19 - 8 = 11
    11 is divisible by 11. Hence 2737152 is also divisible by 11.

    Example3: Check if 957 is divisible by 11.

    9 + 7 = 16
    5 = 5
    16 - 5 = 11
    11 is divisible by 11. Hence 957 is also divisible by 11.

    Example4: Check if 9548 is divisible by 11.
    9 + 4 = 13
    5 + 8 = 13
    13 - 13 = 0
    We got the difference as 0. Hence 9548 is divisible by 11.
    Divisibility by 12A number is divisible by 12 if the number is divisible by both 3 and 4Example1: Check if 720 is divisible by 12.

    720 is divisible by 3 and 720 is also divisible by 4. (Check thedivisibility rules of 3 and 4 to find out this)

    Hence 720 is also divisible by 12

    Example2: Check if 916 is divisible by 12.

    916 is not divisible by 3 , though 916 is divisible by 4.(Check thedivisibility rules of 3 and 4 to find out this)

    Hence 916 is not divisible by 12

    Example3: Check if 921 is divisible by 12.

    921 is divisible by 3. But 921 is not divisible by 4.(Check the divisibility rules of 3 and 4 to find out this)

    Hence 921 is not divisible by 12

    Example4: Check if 827 is divisible by 12.

    827 is not divisible by 3. 827 is also not divisible by 4.(Check thedivisibility rules of 3 and 4 to find out this)

    Hence 827 is not divisible by 12
    Divisibility by 13To find out if a number is divisible by 13, multiply the last digit by 4 and add it to the number formed by the remaining digits.

    Repeat this process until we get at a smaller number whose divisibility we know.

    If this smaller number is divisible by 13, the original number is also divisible by 13.    		Example1: Check if 349 is divisible by 13.

    Given number = 349
    34 + (9 × 4) = 34 + 36 = 70
    70 is not divisible by 13. Hence 349 is not divisible by 349

    Example2: Check if 572 is divisible by 13.

    Given number = 572
    57 + (2 × 4) = 57 + 8 = 65
    65 is divisible by 13. Hence 572 is also divisible by 13

    Example3: Check if 68172 is divisible by 13.

    Given number = 68172
    6817 + (2 × 4) = 6817 + 8 = 6825
    682 + (5 × 4) = 682 + 20 = 702
    70 + (2 × 4) = 70 + 8 = 78
    78 is divisible by 13. Hence 68172 is also divisible by 13.

    Example4: Check if 651 is divisible by 13.

    Given number = 651
    65 + (1 × 4) = 65 + 4 = 69
    69 is not divisible by 13. Hence 651 is not divisible by 13
    Divisibility by 14A number is divisible by 14 if it is divisible by both 2 and 7.Example1: Check if 238 is divisible by 14

    238 is divisible by 2 . 238 is also divisible by 7. (Please check thedivisibility rule of 2 and 7 to find out this)
    Hence 238 is also divisible by 14

    Example2: Check if 336 is divisible by 14

    336 is divisible by 2 . 336 is also divisible by 7. (Please check thedivisibility rule of 2 and 7 to find out this)
    Hence 336 is also divisible by 14

    Example3: Check if 342 is divisible by 14.

    342 is divisible by 2 , but 342 is not divisible by 7.(Please check thedivisibility rule of 2 and 7 to find out this)
    Hence 342 is not divisible by 12

    Example4: Check if 175 is divisible by 14.

    175 is not divisible by 2 , though it is divisible by 7.(Please check thedivisibility rule of 2 and 7 to find out this)
    Hence 175 is not divisible by 14

    Example5: Check if 337 is divisible by 14.

    337 is not divisible by 2 and also by 7 (Please check the divisibility rule of 2 and 7 to find out this)
    Hence 337 is not divisible by 14
    Divisibility by 15A number is divisible by 15 If it is divisible by both 3 and 5.Example1: Check if 435 is divisible by 15

    435 is divisible by 3 . 435 is also divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this)
    Hence 435 is also divisible by 15

    Example2: Check if 555 is divisible by 15

    555 is divisible by 3 . 555 is also divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this)
    Hence 555 is also divisible by 15

    Example3: Check if 483 is divisible by 15.

    483 is divisible by 3 , but 483 is not divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this)
    Hence 483 is not divisible by 15

    Example4: Check if 485 is divisible by 15.

    485 is not divisible by 3 , though it is divisible by 5. (Please check thedivisibility rule of 3 and 5 to find out this)
    Hence 485 is not divisible by 15

    Example5: Check if 487 is divisible by 15.

    487 is not divisible by 3 . It is also not divisible by 5 (Please check thedivisibility rule of 3 and 5 to find out this)
    Hence 487 is not divisible by 15
    Divisibility by 16A number is divisible by 16 if the number formed by the last four digits is divisible by 16.Example1: Check if 5696512 is divisible by 16.

    The number formed by the last four digits of 5696512 = 6512
    6512 is divisible by 16. Hence 5696512 is also divisible by 16.

    Example2: Check if 3326976 is divisible by 16.

    The number formed by the last four digits of 3326976 = 6976
    6976 is divisible by 16. Hence 3326976 is also divisible by 16.

    Example3: Check if 732374360 is divisible by 16.

    The number formed by the last three digits of 732374360 = 4360
    4360 is not divisible by 16. Hence 732374360 is also not divisible by 16.
    Divisibility by 17
    To find out if a number is divisible by 17, multiply the last digit by 5 and subtract it from the number formed by the remaining digits.
    Repeat this process until you arrive at a smaller number whose divisibility you know.
    If this smaller number is divisible by 17, the original number is also divisible by 17.
    		Example1: Check if 500327 is divisible by 17.

    Given Number = 500327
    50032 - (7 × 5 )= 50032 - 35 = 49997
    4999 - (7 × 5 ) = 4999 - 35 = 4964
    496 - (4 × 5 ) = 496 - 20 = 476
    47 - (6 × 5 ) = 47 - 30 = 17
    17 is divisible by 17. Hence 500327 is also divisible by 17

    Example2: Check if 521461 is divisible by 17.

    Given Number = 521461
    52146 - (1 × 5 )= 52146 -5 = 52141
    5214 - (1 × 5 ) = 5214 - 5 = 5209
    520 - (9 × 5 ) = 520 - 45 = 475
    47 - (5 × 5 ) = 47 - 25 = 22
    22 is not divisible by 17. Hence 521461 is not divisible by 17
    Divisibility by 18
    A number is divisible by 18 if it is divisible by both 2 and 9.
    		Example1: Check if 31104 is divisible by 18.

    31104 is divisible by 2. 31104 is also divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this)
    Hence 31104 is divisible by 18

    Example2: Check if 1170 is divisible by 18.

    1170 is divisible by 2. 1170 is also divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this)
    Hence 1170 is divisible by 18

    Example3: Check if 1182 is divisible by 18.

    1182 is divisible by 2 , but 1182 is not divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this)
    Hence 1182 is not divisible by 18

    Example4: Check if 1287 is divisible by 18.

    1287 is not divisible by 2 though it is divisible by 9. (Please check thedivisibility rule of 2 and 9 to find out this)
    Hence 1287 is not divisible by 18
    Divisibility by 19
    To find out if a number is divisible by 19, multiply the last digit by 2 and add it to the number formed by the remaining digits.
    Repeat this process until you arrive at a smaller number whose divisibility you know.
    If this smaller number is divisible by 19, the original number is also divisible by 19.
    		Example1: Check if 74689 is divisible by 19.

    Given Number = 74689
    7468 + (9 × 2 )= 7468 + 18 = 7486
    748 + (6 × 2 ) = 748 + 12 = 760
    76 + (0 × 2 ) = 76 + 0 = 76
    76 is divisible by 19. Hence 74689 is also divisible by 19

    Example2: Check if 71234 is divisible by 19.

    Given Number = 71234
    7123 + (4 × 2 )= 7123 + 8 = 7131
    713 + (1 × 2 )= 713 + 2 = 715
    71 + (5 × 2 )= 71 + 10 = 81
    81 is not divisible by 19. Hence 71234 is not divisible by 19
    Divisibility by 20A number is divisible by 20 if it is divisible by 10 and the tens digit is even.

    (There is one more rule to see if a number is divisible by 20 which is given below.
    A number is divisible by 20 if the number is divisible by both 4 and 5)    		Example1: Check if 720 is divisible by 20

    720 is divisible by 10. (Please check the divisibility rule of 10 to find out this).
    The tens digit = 2 = even digit.
    Hence 720 is also divisible by 20

    Example2: Check if 1340 is divisible by 20

    1340 is divisible by 10. (Please check the divisibility rule of 10 to find out this).
    The tens digit = 2 = even digit.
    Hence 1340 is divisible by 20

    Example3: Check if 1350 is divisible by 20

    1350 is divisible by 10. (Please check the divisibility rule of 10 to find out this).
    But the tens digit = 5 = not an even digit.
    Hence 1350 is not divisible by 20

    Example4: Check if 1325 is divisible by 20

    1325 is not divisible by 10 (Please check the divisibility rule of 10 to find out this) though the tens digit = 2 = even digit.
    Hence 1325 is not divisible by 20

  • What are Factors of a Number and how to find it out?   


    • Factors of a number
      If one number is divisible by a second number, the second number is a factor of the first number
      The lowest factor of any positive number = 1
      The highest factor of any positive number = the number itself
      Example
      The factors of 36 are 1, 2, 3, 4, 6, 9 12, 18, 36 because each of these numbers divides 36 with a remainder of 0
    • How to find out factors of a number
      • Write down 1 and the number itself (lowest and highest factors).
      • Check if the given number is divisible by 2 (Reference: Divisibility by 2 rule)
        If the number is divisible by 2, write down 2 as the second lowest factor and divide the given number by 2 to get the second highest factor
      • Check for divisibility by 3, 4,5, and so on. till the beginning of the list reaches the end
      Example1: Find out the factors of 72
      Example2: Find out the factors of 22
    • Important Properties of Factors
      • If a number is divisible by another number, then it is also divisible by all the factors of that number.

      Example : 108 is divisible by 36 because 106 ÷ 38 = 3 with remainder of 0.

      The factors of 36 are 1, 2, 3, 4, 6, 9 12, 18, 36 because each of these numbers divides 36 with a remainder of 0.

      Hence, 108 is also divisible by each of the numbers 1, 2, 3, 4, 6, 9, 12, 18, 36.

  • What are Prime Numbers and Composite Numbers?   


    • Prime Numbers
      • A prime number is a positve integer that is divisible by itself and 1 only. Prime numbers will have exactly two integer factors.
      Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.
      Please note the following facts
      Zero is not a prime number because zero is divisible by more than two factors. Zero can be divided by 1, 2, 3 etc.
      (0 ÷ 1 = 0, 0÷ 2 = 0 ...)
      One is not a prime number because it does not have two factors. It is divisible by only 1
    • Composite Numbers
      • Composite numbers are numbers that have more than two factors. A composite number isdivisible by at least one number other than 1 and itself.
      Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc.
      Please note that zero and 1 are neither prime numbers nor composite numbers.
      Every whole number is either prime or composite, with two exceptions 0 and 1 which are neither prime nor composite

  • 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   Minimize Button


    • 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 andhow 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 ?    Minimize Button


    • 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 Minimize Button


    • 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

  • 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
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