HCF & LCM of Numbers

 HCF & LCM

If a number 'a' divides another number 'b' exactly,  then we can say 'a' is a factor of 'b' and 'b' is called a multiple of 'a'.

HCF (Highest Common Factor) or GCM( Greatest Common Measure) or GCD (Greatest Common Divisor) :- HCF of two or more than two numbers is the greatest number that divides each of them exactly. 

LCM (Least Common Multiple) :- The least number which is divisible by two or more given numbers, then that least number is called LCM of the numbers. 

Methods for calculating HCF & LCM

1. Factor Method
2. Division Method 

• Product of two numbers = Product of their HCF and LCM 

HCF and LCM of fractions  :- 

1. HCF = HCF of Numerators/LCM of Denominators
2. LCM = LCM of Numerators/HCF of Denominators

• If there is no common factor between two numbers,  then LCM will be the product of both the numbers and HCF is equal to 1.

Example 

1. Find the HCF of 513, 1134 and 1215?
Solution:
     By Division Method, 
    HCF of 1134 and 1215 = 81
    Then, HCF of 513 and 81 = 27
    So, answer is 27.

2. Find HCF and LCM of 2³ × 3³ × 5 × 7², 2 × 3² × 5³ × 7, 2² × 3 × 5³ × 7³ ?
Solution:
HCF = 2 × 3 × 5 × 7 = 210
LCM = 2³ × 3³ × 5³ × 7³ 

3. Find HCF and LCM of 2/3, 3/5, 4/7, 9/13 ?
Solution:
HCF = HCF(2, 3, 4, 9)/LCM(3, 5, 7, 13) = 1/1365
LCM = LCM (2, 3, 4, 9)/HCF(3, 5, 7, 13) = 36

4. Find the largest number which divides 62, 132 and 237 to leave the same remainder in each case?
Solution:
Tricks  - In case of same remainder,  each number is subtracted from each other  and their HCF is the answer. 
So, HCF[(132 - 62), (237 - 132), (237 - 62)] = 35

5. The HCF and LCM of two numbers are 21 and 84 respectively. If the ratio of two numbers is 1 : 4, then the larger number is ----?
Solution:
Let the numbers be x and 4x.
x × 4x = 84 × 21 , then x = 21
The larger number = 4x = 4 × 21 = 84

6. The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively is ---?
Solution:
1657 - 6 = 1651 and 2037 - 5 = 2032
HCF (1651, 2032) = 127

7. Three numbers which co-primes to each other are such that the product of the first two is 551 and that of last two is 1073. The sum of the three numbers is-----?

Solution:

Since, the numbers are co-primes,  they contain only 1 as common factor.  Also,  the given two products have the middle number in common. 
So, middle number = HCF (551, 1073) = 29
First number = 551/29 = 19
Third number = 1073/29 = 37
Required sum = 19 + 29 + 37 = 85

Previous Year Questions

1. The LCM of a and b is 42. The LCM of 5a and 11b is ---------? (Ans.2310)

2. The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm & 12 m 95 cm is --------? (Ans.35 cm)

3. The least number which should be added to 2497, so that the sum is exactly divisible by 5, 6, 4 and 3 is -----? (Ans. 23)

4. The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is ------? ( Ans.364)

5. The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder,  is -----? (Ans.1683)

6. If the sum of two numbers is 55 and the HCF and LCM of these numbers are 5 and 120 respectively,  then the sum of the reciprocals of the numbers is equal to .....? ( Ans.11/120)