Concept of LCM & HCF and related questions for government job exams

LCM and HCF topic is a part and parcel of Quantitative Ability test, asked in almost every competitive exams. In Government job exams like SSC CGL, SSC CHSL, IBPS PO and Clerk, SBI or RBI etc, surely asks 1-2 question on this topic and which can be solved in seconds or within a minute, if prior concept have known and practice is done.
So, i have elaborated the concept of LCM and HCF and also written down the questions that are usually asked in these types of exams.

Concept of LCM and HCF:

1. LCM (Least Common Multiple): The least number which is exactly divisible by each one of the number given. For example: LCM of 2,4,5 is 20.

2. HCF (Highest Common Factor) or GCD (Greatest Common Divisor): It is the greatest number that divides the each of the number given. For example: HCF of 2,4,8 is 2 or HCF of 2,4,5 is 1 (here, 1 is the only number that divides all the given number i.e. 2,4,5)

Note: If a number, let A divides the other number, let B exactly, i.e. B/A that means A is a factor of B and B is a multiple of A.

3. Co-primes: The two numbers are said to co-primes when there HCF is 1.

4. The product of 2 numbers = the product of LCM and HCF
N1 x N2 = LCM x HCF

5. HCF of fractions (N1/N2) = HCF of Numerator/LCM of denominator
– LCM of fractions = LCM of Numerator/HCF of denominator

Questions on LCM and HCF for SSC and Bank exams:

1. Find the HCF of 204, 1190 and 1445

For larger numbers- The HCF of 2923 and 3239

2. Find the LCM of 22, 54, 108, 135 and 198

3. Find the HCF and LCM of 9/10, 12/25, 18/35 and 21/40
– Using the Formula of HCF of Fractions = HCF of numerator/LCM of denominator
   => HCF of 9, 12, 18 and 21/ LCM of 10, 25, 35, 40 = 3/2800
– Using the formula of LCM of Fractions = LCM of numerator/HCF of denominator
  => LCM of 9, 12, 18, 21/ HCF of 10, 25, 35, 40 = 252/5 
4. Find the HCF and LCM of 1.75, 7 and 5.6
– We can write the numbers in fractions i.e. 175/10, 7/1 and 56/10
Now, using HCF and LCM of Fractions => HCF of fractions = 7/10 and LCM of fractions = 1400/1
5. The HCF and LCM of 2 numbers are 18 and 108 respectively and one number is 36. Find the other number?
– Using formula = HCFxLCM = N1xN2 => 18×108 = 36xN2 => N2 = 54
6. The HCF of 3240, 3600 and a third number is 36 and LCM is 24 x 35 x 52 x 72. Find the third number?
– 3240 = 23 x 34 x 5; 3600 = 24 x 32 x 52; HCF = 36 = 22 x 32
Since, HCF is the product of lowest common factors, so third number must have (22 x 32) as its factor.
Since, LCM is the product of highest common factors, so third number must have (35 x 72) as its factor
So, third number =  22 x 35 x 72.
7. Three numbers are in the ratio of 1:2:3 and their HCF is 12. Find the respective numbers and their LCM?

8. The product of 2 numbers is 2028 and their HCF is 13. Find the number of such pairs?

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

10. The HCF and LCM of two numbers are 21 and 84 respectively. If the ratio of 2 numbers is 1:4, find the numbers, their sum and their difference?

11. The LCM of two numbers is 495 and their HCF is 5. If the sum of numbers is 10, find their difference?

12. The greatest possible length which can be used to measure exactly 7m, 560cm and 910cm.
– Here, we have to find the HCF of 7m, 560cm and 910cm.
 HCF of 700cm, 560cm and 910cm = 70

13. The maximum number of students among them 1001 pens and 910 pencils can be distributed in such a way that each student get the same number of pens and pencils.
– Here, we have to find the HCF so that least number of pens and pencils can be calculated which can be equally distributed among the students. Hence, the HCF of 1001 and 910 = 91

14. Let N be the greatest number that can divide 1305, 4665 and 6905 leaving the same remainder in each case. Find the number N and its sum?

15. The least number when divided by 4, 6, 10 and 15 leaves remainder 2 in each case. Find the number?
– The LCM of 4,6,10 and 15 = 60
and the number will be 60+2 (2 is the remainder) = 62
16. The least number which should be added to 2497 so that sum is exactly divisible  by 5, 6, 4 and 3 is?

17. Find the least multiple of 23, when divided by 18, 21 and 24 leaves remainder 7, 10 and 13 respectively.

18. Find the largest 4-digit number which when divided by 4, 7 or 13 leaves a remainder 3 in each case?

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