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.

Least Common Multiple Using Prime Factorization

LEAST COMMON MULTIPLE:

Multiple:

A number is said to be multiple of another number, when it is exactly divisible by other number.

Example:  10 is multiple of 2 and 5.

Common Multiple:

Common Multiple of two or more numbers is a  number which is exactly divisible by each of them.

Example: 12 is a common multiple of 2, 3, 4, 6.

 

Least Common Multiple:

Least Common Multiple(L.C.M) is also called as Smallest Common Multiple or Smallest Common Divisor. 

The least number exactly divisible by each one of the given numbers is called least common multiple.

The Least Common Multiple of Numbers can be found out by any one of the methods:

(i) L.C.M using Prime Factorization Method.

(ii) L.C.M using Common Division Method.

(iii) L.C.M using Listing out Multiples Method.

By using any one of the above methods in finding L.C.M gives same L.C.M.

 

(i) L.C.M of Numbers Using Prime Factorization Method:

Step-1: We need to express the given numbers in terms of their Prime factors either using Factor-tree method or Short division method.

Step-2: Check for the common prime factors and find the highest index of each common prime factor.

Step-3: The product of all Prime factors and common Prime factors with respective to highest indices is the Least Common Multiple of the given numbers.

 

Find L.C.M of numbers:

1)  12 and 8.

Answer:

Step-1: Factors of  12 and 8

Prime Factors of 12 = 2 * 2 * 3 = 2² * 3.

Prime Factors of 8 = 2 * 2* 2 = 2³.

Step-2: The common factors of both numbers are 2.The highest index of prime factor 2 is 3. Other prime factor is 3.
Step-3:  Product of all Prime factors and the common prime factor with highest idex gives L.C.M.

Therefore, L.C.M of 12 and 8 = 2³ * 3 = 8 * 3 = 24.

 

2) 3, 13, 33

Answer:

Step-1: Factors of 3, 13, 33.

As, given numbers 3 and 13 are  Prime numbers. So, the factors of 3 and 13 are number itself.

Prime Factors of  3 = 3.

Prime Factors of 13 = 13.

Prime Factors of 33 = 3 * 11.   

Step-2: The common Prime factors of the given numbers =3. Other Prime factors are = 13, 11.

Step-3: Product of all the prime factors and the common prime factor gives L.C.M.

Therefore, L.C.M of 3, 13 and 33 = 3 * 13 * 11 = 429.  

 

3) 16, 24, 40.

Answer:

Step-1: Factors of the given numbers:

Prime Factors of 16 =  2 * 2 * 2 * 2 = 2⁴.

Prime Factors of 24 = 2 * 2 * 2 *3 = 2³ * 3.

Prime Factors of 40 =  2* 2* 2 * 5 = 2³ * 5.

Step-2: The Common Prime factor of the given numbers 2,  with the highest idex is 4. i.e: 2⁴. The other Prime factors are :  3 and 5.

Step-3: The Product of all the Prime factors gives L.C.M of the given numbers.

Therefore, L.C.M of 16, 24, 40 = 2⁴ * 3 * 5 = 16 * 3 * 5 = 240.

 

4) 27, 36, 90.

Answer:

Step-1: Factors of the given numbers:

Prime Factors of 27 = 3 * 3 * 3 = 3³.

Prime Factors of 36 = 2 * 2 * 3 * 3 = 2² * 3².

Prime Factors of 90 = 2 * 3 * 3 * 5 = 2 * 3² * 5.

Step-2: The Common prime factors of the given numbers with highest index = 2² , 3³ . The other prime factor is 5.

Step-3: The product of all the prime factors and the common prime factors with highest index gives L.C.M.

Therefore, L.C.M of 27, 36 , 90 = 2² * 3³ * 5 = 4 * 27 * 5 = 540.