L.C.M of Decimal Numbers.

LCM of Decimals:

We need to make the same number of decimal places in all the given numbers; then find their LCM as if they were integers, and mark in the result as many decimal places as there are in each of the numbers.

1) 0.6, 9.6 and 0.36.

Answer:

The given numbers are 0.6, 9.6 and 0.36
(i) Converting the Decimal numbers to Like Decimals= 0.60 , 9.60 , 0.36

 Now, find the LCM of 60, 960 and 36 = 2880.

 

(ii)The decimal point is to be placed for the L.C.M.
∴ The required LCM = 28.80.

                         (or)

(i) Convert the given decimal numbers into Fractions:

0.6 = 6/10 ; 9.6 = 96/10 ; 0.36 = 36/100.

L.C.M ( x/a , y/b, z/c ) = L.C.M (x,y,z) / H.C.F ( a,b,c ).

Therefore, L.C.M( 6/10 , 96/10 , 36/100 )

= L.C.M(6,96,36) / H.C.F(10,10,100)

(ii) L.C.M(6, 96, 36) =  288.

      H.C.F(10,10,100) = 10.

(iii) L.C.M( 6/10 , 96/10 , 36/100 )

       =  288 / 10 = 28.8

 

2) 0.48, 0.72 and 0.108
Answer:

Given decimal numbers: 0.48, 0.72 and 0.108
Converting each of the following decimals into like decimals we get;
0.480, 0.720 and 0.108
Now, expressing each of the numbers without the decimals as the product of primes we get
480 = 2 × 2 × 2 × 2 × 2 × 3 × 5 = 2⁵ × 3 × 5
720 = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 2⁴ × 3² × 5
108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³
L.C.M. of 480, 720 and 108 = 2⁵ × 3³ × 5 = 4320
Therefore, L.C.M. of 0.48, 0.72, 0.108 = 4.32 (taking 3 decimal places)

                     (or)

(i) Convert the given decimal numbers into Fractions:

0.48 = 48/100 ; 0.72 = 72/100 ; 0.108 = 108/1000.

L.C.M ( x/a , y/b, z/c ) = L.C.M (x,y,z) / H.C.F ( a,b,c ).

Therefore, L.C.M( 48/100 , 72/100 , 108/1000 )

= L.C.M(48, 72, 108) / H.C.F(100,100,1000)

(ii) L.C.M(48, 72, 108) =  432.

      H.C.F(100,100,1000) = 100.

 

Therefore, L.C.M( 48/100 , 72/100 , 108/1000 )

= 432 / 100 = 4.32 .

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

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

a) H.C.F = H.C.F  of Numerator

                  L.C.M of Denominator

H.C.F ( x/a,  y/b,  z/c ) =  H.C.F(x, y, z) / L.C.M( a,b,c) .

 

L.C.M ( x/a,  y/b,  z/c) =  L.C.M (x,y,z) / H.C.F(a,b,c).    

 

Find the H.C.F and L.C.M of Fractions:

1) 2/4,  5/6,  10/8.

Answer:

a) H.C.F( 2/4, 5/6, 10/8) = H.C.F( 2, 5, 10)

                                        L.C.M( 4, 6, 8)

 

We need to find H.C.F and L.C.M of numbers.

 

H.C.F: Listing out Factors Method	L.C.M: Listing out Multiples      Method
Factors of 2 = 1, 2. Factors of 5 = 1, 5. Factors of 10 = 1, 2, 5, 10. Common Factors( 2, 5, 10) = 1.         Therefore, H.C.F( 2, 5, 10) = 1.	Multiples of 4 = 4,8,16,20,24,28,32,36,40. Multiples of 6 = 6,12,18,24,30,36,42,48,54,60. Multiples of 8 = 8,16,24,32,40,48,56,64,72,80. L.C.M(4, 6, 8) = 24.

 

H.C.F( 2/4, 5/6, 10/8) = H.C.F( 2, 5, 10)

                                        L.C.M( 4, 6, 8)

 

                                    =   1  .

                                        24 

 

b) L.C.M ( 2/4, 5/6, 10/8) =     L.C.M( 2, 5, 10)

                                                   H.C.F( 4, 6, 8)

 

H.C.F: Listing out Factors Method	L.C.M: Listing out Multiples      Method
Factors of 4 = 1, 2, 4. Factors of 6 = 1, 2, 3, 6. Factors of 8 = 1, 2, 4, 8. Common Factors( 4, 6, 8) = 1,2.          Therefore, H.C.F( 4, 6, 8) = 2.	Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. L.C.M(2, 5, 10) = 10.

 

L.C.M ( 2/4, 5/6, 10/8) =     L.C.M( 2, 5, 10)

                                               H.C.F( 4, 6, 8)

 

                                       =  10  =  5.

                                            2

L.C.M using Listing Out Multiples Method.

L.C.M Using Listing Out Multiples Method:

The Steps to find L.C.M by using Listing Out Multiples method are:

(i) We need to find out the Multiples of the given numbers. First write the first 10 multiples of the given numbers.If the common multiple is not found, the write up to next 10 multiples. Repeat the multiples till common multiple is found.  

(ii)  The Common Multiples of the numbers are taken out.

(iii)  The Least or Smallest Multiple in those Common Multiples is the L.C.M.

 

Find the L.C.M of numbers using Listing out multiples:

1) 3 , 4.

Answer:

Step-1: Multiples of the given numbers:

Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40.

Step-2: The common multiples of the given numbers are = 12, 24.

Step-3: The Smallest Common Multiple in 12 and 24 is 12.

Therefore, L.C.M of 3, 4 = 12.

 

2) 8, 7.

Answer:  

Step-1: Multiples of the given numbers:

Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.

Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70.

Step-2: The common multiples of the given numbers for first 10 multiples= 56.

Step-3: The Smallest Common Multiple is 56.

Therefore, L.C.M of  8, 7 =  56.

 

3) 2, 8, 10.

Answer:

Step-1: Multiples of the given numbers:

Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40.

Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80.

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

Step-2: The common multiples of the given numbers = 40.

Step-3: The Smallest Common Multiple is 40.

Therefore, L.C.M of  2, 8, 10 =  40.

 

4) 10, 20 25.

Answer:

Step-1: Multiples of the given numbers:

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.

Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200.

Multiples of 25 = 25, 50, 75, 100, 125, 150, 175,  200, 225, 250.

Step-2: The common multiples of the given numbers= 100.

Step-3: The Smallest Common Multiple is 100.

Therefore, L.C.M of 10, 20, 25 = 100.

L.C.M using Common Division Method.

 L.C.M Using Common Division Method:

Step-1: Divide the given numbers with their common prime factors.

Step-2: Divide till the given numbers have no common prime factors.

Step-3: Finally multiply the common prime factors and the remainders to obtain L.C.M of the given numbers.

 

Find L.C.M of numbers using Common Division method:

1) 86, 68, 20.

Answer:

Step-1: Divide the given numbers with their common prime factors.

Step-2: Common Prime factors= 2 ,2.

              Remainders = 43, 17, 5.

Step-3: Multiply the Common Prime Factors and the remainders to obtain L.C.M of the given numbers.

Therefore, L.C.M of 86, 68, 20 = Common Prime factors * Remainders = (2 * 2 )(43 * 17 * 5) = 4 * 43 * 85 = 14620.

 

2) 102, 170, 136.

Answer:

Step-1: Divide the given numbers with their common prime factors.

Step-2: Common Prime Factors = 17, 2, 2.

              Remainders = 3, 5, 2.

Step-3: Multiply the Common Prime Factors and the remainders to obtain L.C.M of the given numbers.

Therefore, L.C.M of 102, 170, 136 = Common Prime factors * Remainders = (17* 2* 2)(3 * 5 * 2) = 68 * 30 = 2040.  

 

3) 36, 18, 9.

Answer:

Step-1: Divide the given numbers with their Common Prime Factors.

Step-2: Common Prime Factors = 3, 3, 2.

              Remainder = 2.

Step-3: Multiply the Common Prime Factors and the remainders to obtain L.C.M of the given numbers.

Therefore, L.C.M of 36, 18, 9 = Common Prime Factors * Remainders = (3 * 3 * 2)(2) = 18 * 2 = 36.

4) 120, 150, 135.

Answer:

Step-1: Divide the given numbers with their Common Prime Factors.

Step-2: Common Prime Factors = 5, 3, 2, 2, 3.

              Remainders = 2, 5, 3.

Step-3: Multiply the Common Prime Factors and the remainders to obtain L.C.M of the given numbers.

Therefore, L.C.M of 120, 150, 135 = Common Prime Factors * Remainders = (5 * 3 * 2* 2 * 3)(2 * 5 * 3) = 180 * 30 = 5400.