LCM & HCF
LCM & HCF
(1) Find the HCF and LCM of the Following Numbers by Listing Just the Multiples of One of Them
(a)  Number  Multiples of one of the numbers  LCM  Factors  HCF 
6036  
(b)  88144  
(c)  132420 
(2) Find the HCF and LCM of 144 and 180 by Finding Their Prime Factors
HCF
LCM
(3) Find the HCF and LCM of the Following Numbers by Expressing Them Using Exponential Notation of the Prime Factored Form
(a) 180 and 756
LCM
HCF
(b) 66 and 88
LCM
HCF
(4) Find the HCF and LCM of the Following Numbers Using the Common Division Method
(a) 12,15 and 21
LCM
HCF
(5) Find the HCF and the LCM of the Following Numbers Using the Common Division Method
(a) 50,80 and 160
LCM
HCF
(6) Find the HCF and the LCM of the Following Numbers Using the Common Division Method
(a) 48,72 and 54
LCM
HCF
(7) if a and B Are Two Numbers, Fill in the Missing Boxes
A  B  LCM (A, B)  HCF (A, B)  
(a)  
(b)  
(c) 
(8) Answer the Following Questions
(a) Three bells ring at intervals of seconds respectively. If they ring together now, after how many seconds will all the bells ring together again?
(b) Vinay is buying nuts and bolts at a hardware store. The store sells nuts in packs of 20 and bolts in packs of 5. If Vinay wishes to buy the same number of nuts and bolts, what is the smallest number of nuts he can buy?
Answers and Explanations
Answer (1)
 The least number which is exactly divisible by each of the given number is called the least common multiple of those numbers.
 For example: consider the numbers 3,31 and .
 The LCM of these numbers would be
 The largest number that divides two or more numbers is the highest common factor for those numbers. For example: consider the numbers
 3 is the largest number that divides each of these numbers, and hence HCF is 3.
 Now, below table represent the HCF and LCM:
(a)  Number  Multiples of one of the numbers  LCM  Factors  HCF 
6036  
(b)  88144  


(c)  132420 



Answer (2)
 The least number which is exactly divisible by each of the given number is called the least common multiple of those numbers.
 The largest number that divides two or more numbers is the highest common factor for those numbers.
 Now, we can find prime factorization of both 144 and 180:
 So,
 Now, HCF (highest common factors)
 Then, LCM (Least common multiple)
 Therefore, the answer is
Answer 3 (A)
 Here, given number 180 and 756
 Now, first find HCF:
 HCF
 Now, Exponential form
 Now, find LCM:
 LCM
 Now, Exponential form
 Therefore, the answer is
Answer 3 (B)
 Here, given number 66 and 88
 Now, first find HCF:
 HCF
 Now, Exponential form
 Now, find LCM:
 LCM
 Now, Exponential form
 Therefore, the answer is
Answer 4 (A)
 Here, given number 12,15 and 21
 Now, we can find HCF:
12
15
21
 HCF: Product of smallest power of each common factor
 Now, find LCM:
 Product of greatest power of each prime factor
 Therefore, the answer is
Answer 5 (A)
 Here, given number 50,80 and 160
 Now, we can find HCF:
 50
 80
 160
 HCF: Product of smallest power of each common factor
 Now, find LCM:
 Product of greatest power of each prime factor

 Therefore, the answer is
Answer 6 (A)
 Here, given number 48,72 and 54
 Now, we can find HCF:

 48
 72
 54
 HCF: Product of smallest power of each common factor
 Now, find LCM:
 Product of greatest power of each prime factor

 Therefore, the answer is
Answer 7 (A)
 The Least Common Multiple of two integers and , usually denoted by LCM (a, b) is the smallest positive integer that is divisible by both and .
 Check out the LCM formula for any two numbers using HCF in the given below:
 Now, put the value in above formula:
A  B  LCM (A, B)  HCF (A, B)  
(a) 
Answer 7 (B)
 Here, given product of A and B is 1728 and value of B is 32
 So, find the value of A:
 So, the value of A is 54
 Now, to find HCF of 54 and 32:
 54
 32
 Now, HCF (32,54)
A  B  LCM (A, B)  HCF (A, B)  
(b) 
Answer 7 (C)
 The Least Common Multiple of two integers and , usually denoted by LCM (a, b) is the smallest positive integer that is divisible by both and .
 Check out the LCM formula for any two numbers using HCF in the given below:
 Now, put the value in above formula:
 Now, product of A and B:
A  B  LCM (A, B)  HCF (A, B)  
(c) 
Answer 8 (A)
 Three bells ring at intervals of seconds respectively.
 If they ring together now, so these problem based on the concept of LCM.
 Here we have to find LCM of intervals 3,6 and 9 seconds which indicates after how long they all again ring together.
 Write 3,6 and 9 product of prime factors.
 LCM (3,6, 9)
 Therefore, three bells ring together after 18 seconds.
Answer 8 (B)
 Vinay is buying nuts and bolts at a hardware store.
 The store sells nuts in packs of 20 and bolts in packs of 5.
 If Vinay wishes to buy the same number of nuts and bolts,
 Now, find the LCM of nuts and bolts:
 LCM (20,5) :
 LCM (20,5)
 Therefore, he can buy minimum number of 20 nuts.