HCF AND LCM

Factors and Multiples:

If number a divided another number b exactly, we say that a is a factor of b.

In this case, b is called a multiple of a.

Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.):

The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly.

There are two methods of finding the H.C.F. of a given set of numbers:

1. Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

2. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.

   Similarly, the H.C.F. of more than three numbers may be obtained.

Finding the H.C.F. of more than two numbers: 

Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.

Similarly, the H.C.F. of more than three numbers may be obtained.

Least Common Multiple (L.C.M.):

The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

There are two methods of finding the L.C.M. of a given set of numbers:

1. Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

2. Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

Product of two numbers = Product of their H.C.F. and L.C.M.

Coprimes : Two numbers are said to be co-primes if their H.C.F. is 1

HCF and LCM of fractions

1. H.C.F =	H.C.F. of Numerators
	L.C.M. of Denominators

2. L.C.M =.	L.C.M. of Numerators
	H.C.F. of Denominators

H.C.F. and L.C.M. of Decimal Fractions:

In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

Comparison of Fractions:

Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

EXAMPLES

Exp: 1 Find HCF of 6/5, 12/25 and 18/35 .

Sol:

HCF (6/5, 12/25, 18/35) =

HCF(6,12,18) / LCM(5,25,35)

= 6/175 

Here, every fraction is exactly divisible by hcf 6/175. Exactly divisible means the quotient will be an integer, with remainder=0 like

6/5 ÷ 6/175 = (6*175) ÷ (5*6) = quotient= 35, R=0 . We can check with remaining fractions too.

Exp 2: Find LCM of  6/5, 12/25 and 18/35 .

Sol: LCM( 6/5, 12/25, 18/35 ) =

= LCM(6,12,18) / HCF ( 5,25,35)

= 36/5 

Here, 36/5 is exactly divisible by each given fraction. Like 36/5 ÷ 6/5 = (36*5) / (5*6) = quotient = 6, R= 0. We can check with remaining fractions too.

Exp 3: Find LCM and HCF of  1.20 AND 22.5 .

Sol: Given, 1.20 and 22.5

Converting each of the following decimals into like decimals we get;

1.20 and 22.50

Now, expressing each of the numbers without the decimals as the product of primes we get

120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5

2250 = 2 × 3 × 3 × 5 × 5 × 5 = 2 × 3² × 5³

Now, H.C.F. of 120 and 2250 = 2 × 3 × 5 = 30
Therefore, the H.C.F. of 1.20 and 22.5 = 0.30 (taking 2 decimal places)

L.C.M. of 120 and 2250 = 2³ × 3² × 5³ = 9000
Therefore, L.C.M. of 1.20 and 22.5 = 90.00 (taking 2 decimal places)

For any queries Comment:

Thank you...

                                                                                                                         

For GK Updates: The Infinite Studies

Divisibility Rules

Divisible by 2

If the unit digit of the number is even then the number is divisible by 2.

or

Any number whose unit digit is 0, 2, 4, 6 or 8 then the number is divisible by 2.

example: 892, 3648, 7936, 5566914, 22340 etc.

Divisible by 3

If the sum of all digits of a number is divisible by 3, then the number is divisible by 3.

example: 

i) 897121

sum of unit digits 8+9+7+1+2+1=28

since 28 is not divisible by 3, 897121 is not divisible by 3.

ii) 562137

sum of digits 5+6+2+1+3+7=24

since 24 is divisible by 3, 562137 is also divisible by 3.

Divisible by 4

If the last two digits of a number is divisible by 4, then the number is divisible by 4. 

Last two digits of the number can be 00, 04, 08, 12, 16, 20, 24...,96

example: 764, 82540, 369292, 879684, etc.

Divisible by 5

If the last digit of a number is either  5 or 0, then the number is divisible by 5.

example: 110, 265, 33845, etc.

Divisible by 6

If a number is divisible by both 2 and 3, then the number is divisible by 6.

Or

The even number which is divisible 3 is also divisible by 6.

example: 8640, 7656, 105936 etc.

Divisibility by 7

Multiply the digit of unit place by 2 and subtract from the remaining part of that number. If the resultant number is divisible by 7 then the number is divisible by 7.

example:- 

i) 405

The unit digit is 5 and remaining part of the number is 40.

40-5*2=30

since 30 is not divisible by 7, 405 is also not divisible by 7.

ii) 924

The unit digit is 4 and remaining part of the number is 92.

92-4*2=84

7 divides 84, so 924 is also divisible by 7.

Divisibility by 8

If the number formed by the last three digit of the given number is divisible by 8 then the actual number is divisible by 8.

The last three digit of the number can be 000,008,016,032,064,....,096,104,112,.....,992.

example:-

889344

number formed by last three digit of the given number is 344.

Since 344 is divisible by 8.

Hence 889344 is divisible by 8.

Divisibility by 9

If the sum of digits of the given number is divisible by 9, then the number is divisible by 9.

example:

i) 986321

sum of digits is 9+8+6+3+2+1=29

since 29 is not divisible by 9, hence 986321 is also not divisible by 9.

ii) 7648128

sum of digits 7+6+4+8+1+2+8=36

since 36 is divisible by 9, hence 7648128 is also divisible by 9.

Divisibility by 10

If the last digit of the number is zero, then the number is divisible by 10.

example: 70, 1520, 3548520 etc.

For any query comment below.

Thank you.

---------------------------------------

For GK updates visit:  The Infinite Studies

Tricky Methods to find the square of a number quickly ( Vedic math )

What's square of a number?
When we multiply a number with itself the resultant number is called square of that number.

Finding square by adding consecutive odd number:-

1² = 1

2²=1+3=4

3²=1+3+5=9

4²=1+3+5+7=16

5²=1+3+5+7+9=25

....... So on

Finding Square by using a Base:-

Base:- Bases are generally the powers of 10.

i.e 10,100,1000,10000,........ etc

Case_I:(if the number is less then The Base)

Exp:(1)

let the number is 96

   Here the base is 100 

   Here the number is less then from base by 04

  Then ,

             96²= 96-04 | (04)²

                        = 9216
So the answers is 9216

Exp:(2)

Let the number is 992

 Here the base is 1000

 Here the number is less then from base by 008

Then ,

          992²= 992-008|(008)²

                          = 984064
So the answers is 984064

Exp(3):

Let the number is 9975

  Here the base is 10000

  Here the number is less then base by 0025

Then,

         9975²=9975-0025/(0025)²

                             =99500625
So the answers is 99500625

Case_II:(If the number is greater then the Base)

Exp:(1)

Let the number is 17

  Here the base is 10

  And the number is greater than the base by 07

Then,

         17² = 17+7|(7)²

                 = 24|49

                 =289 
( here the base is the 1st power of 10 so 4 is carried towards the left and added with 24)

So the answers is 289

Exp:(2)

Let the number is 115

  Here the base is 100

  And the number is greater then from base by 15

Then, 

          115²= 115+15|(015)²

                          =130225
So the answers is 130225

Exp(3):

Let The number is 1025

   Here the base is 1000

   And the number is greater than base by 25

Then,

         1025²= 1025+25|(025)²

                          = 1050625
So the answers is 1050625

If the unit digit of the number is "5"

Let MN be the number where N is 5

Then ,

            MN² =M*(M+1)|5²

Exp(1):

Let the number is 45

 45²= 4*5|(5)²

          =2025
So the answers is 2025

Exp(2):

Let the number be 95

   95²= 9*10| 5²

            =9025
So the answers is 9025

Exp(3):

Let the number be 125

 125²= 12*13|5²

           =15625
So the answers is 15625



Dwanda yoga / Duplex combination

The Dwandwa Yoga or 'Duplex Combination' can be used for general purpose squaring.
To proceed further we need to know the Dwandwa of certain numbers.

D( a ) = a²
D( ab ) = 2ab
D( abc ) = 2ac + b²
D( abcd ) = 2ad + 2bc
D( abcde ) = 2ae + 2bd + c²
D( abcdef ) = 2af + 2be + 2cd      and so on....

As we can see above, D of any number is the sum of square of the middle number and two times the product of the other pairs.
Square of a number is given by

( ab )² = D( a ) | D( ab ) | D( b )
( abc )² = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
( abcd )² = D(a) | D(ab) | D(abc) | D(abcd) | D(bcd) | D(bc) | D (c)

Exp(1):
Let the number is 23
Then,

      (23)² = (ab)² = D(a) | D(ab) | D(b)
                               = 4     |    12    |    9
Since Dwanada must have only one digit, we carry over '1' of '12' to LHS.
Therefore it becomes     4 |₁ 2| 9
So the  answer is 529

Exp(2):
Let the number is 527
Then,

    (527)² = ( abc )²
           = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
           =   25   |     20    |     74   |    28    |   49
           =   25   |₂     0    |₇     4   |₂    8    |₄   9
          =   277729
So the answer is 277729

For any query Comment below:-
Thank you.....
_________________________________________________
For GK Updates:  The Infinite Studies