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.    

H.C.F of Three Numbers Using Long Division Method

H.C.F for Three numbers by using Long Division Method:

Step-1: First, we need to find H.C.F of any two numbers.

Step-2: Now, we need to find out the H.C.F of the third number and the H.C.F obtained in step-1.

Step-3: H.C.F obtained in step-2  will be the H.C.F of the three numbers.

 In a similar way by following the steps in finding the H.C.F for Three numbers, we can find out H.C.F for more than three numbers by using Long Division method.

Find the H.C.F of numbers by Long Division method:

1) 255, 136, 170.

Answer:

(i) H.C.F  of  two numbers 136, 170 = 34.

(ii) Now, we need to find out H.C.F of third number:255, and the H.C.F obtained in Step-1 : 34.

i.e; Need to find H.C.F for 34 and 255.

(iii) H.C.F obtained in step-2 is the H.C.F of the three numbers.

Therefore, H.C.F of 255, 136, 170 = 17.

 

2) 144, 252, 630.

Answer:

(i) H.C.F of 144 and 252 = 36.

(ii) Now, we need to find out H.C.F of third number:630, and the H.C.F obtained in Step-1 :36 .

i.e; Need to find H.C.F for 36 and 630.

 

(iii) H.C.F obtained in step-2 is the H.C.F of the three numbers.

Therefore, H.C.F of 144, 252, 630 = 18.

 

3) 639, 1065, 1491.

Answer:

(i) H.C.F  of 1065 and 1491 = 213.

 

(ii) Now, we need to find out H.C.F of third number:639, and the H.C.F obtained in Step-1: 213 .

i.e; Need to find H.C.F for 213 and 639.

(iii) H.C.F obtained in step-2 is the H.C.F of the three numbers.

Therefore, H.C.F of 639, 1065, 1491 = 213.

 

4) 144, 180, 192.

Answer:

(i) H.C.F  of 180, 192 =  12.

(ii) Now, we need to find out H.C.F of third number: 144, and the H.C.F obtained in Step-1: 12.

i.e; Need to find H.C.F for 12 and 144.

(iii) H.C.F obtained in step-2 is the H.C.F of the three numbers.

Therefore, H.C.F of 144, 180, 192 = 12.

H.C.F of Numbers Using Common Division Method

H.C.F of Numbers Using Common  Division Method:

Step-1: Divide the given numbers together by the smallest prime number which can exactly divides all the given numbers.

Step-2: Repeat the process until we reach a stage where no common prime factor exists for all the numbers.

Step-3: We can see that the factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors.

Step-4: Multiply those common prime factors to get the H.C.F.

 

Find the H.C.F of numbers by common division method:

1) 24, 36 and 60.

Answer:

The left side Prime factors which divides the given numbers exactly are 2, 2, 3.

Therefore, H.C.F of 24,36 and 60 = 2 * 2 * 3 = 12.

 

2) 136 ,68, 102.

Answer:

The left side Prime factors which divides the given numbers exactly are 2, 17.

Therefore, H.C.F of 136, 68 , 102 = 2 * 17 = 34.

3) 43, 91, 183.

Answer:

All the given numbers are Prime numbers. So, the factors of Prime numbers are 1 and number itself.

Therefore, H.C.F of 43, 91, 183 = 1.

 

4) 48, 56, 72.

Answer:

The left side Prime factors which divides the given numbers exactly are 2, 2, 2

Therefore, H.C.F of 48, 56, 72 = 2 * 2 * 2 = 8.

 

5) 44, 66,110.

Answer:

The left side Prime factors which divides the given numbers exactly are 2, 11.

Therefore, H.C.F of 44, 66, 110 = 2  * 11 = 22.