HCF and LCM Notes for SSC Exams

HCF and LCM is a high-scoring chapter of SSC Quantitative Aptitude. One or two questions come from it in each Tier 1 paper, and it is needed in fractions, simplification and word problems. These HCF and LCM notes for SSC explain the methods, formulas and standard question types with solved examples.

What Are HCF and LCM?

• HCF (Highest Common Factor) or GCD is the greatest number that divides all the given numbers exactly.
• LCM (Least Common Multiple) is the smallest number that is exactly divisible by all the given numbers.
• HCF is always less than or equal to the smallest number, and LCM is always greater than or equal to the largest number.
• HCF of co-prime numbers is 1, and their LCM is their product.

Methods to Find HCF and LCM

• Prime factorisation: HCF takes the lowest power of each common prime, and LCM takes the highest power of every prime.
• Division method for HCF: divide the larger number by the smaller, then the divisor by the remainder, and repeat until the remainder is zero. The last divisor is the HCF.
• Common division method for LCM: divide the numbers by common primes until no further division is possible, then multiply all divisors and remaining numbers.

Important Formulas

Types of Questions in SSC

• Direct HCF and LCM of given numbers.
• Finding the second number when HCF, LCM and one number are given.
• Greatest number that divides given numbers leaving given remainders.
• Least number that leaves a given remainder when divided by given numbers.
• Bells, lights and traffic signals that come together again after a time.
• Largest tile or the greatest length that measures given lengths exactly.

Solved Examples

Example 1: Find the HCF and LCM of 12 and 18.
Solution: 12 = 2 squared x 3, 18 = 2 x 3 squared. HCF = 2 x 3 = 6. LCM = 2 squared x 3 squared = 36.

Example 2: The HCF of two numbers is 6 and their LCM is 84. If one number is 12, find the other.
Solution: Other number = HCF x LCM / given number = 6 x 84 / 12 = 42.

Example 3: Find the greatest number that divides 60, 84 and 108 exactly.
Solution: HCF of 60, 84 and 108 = 12.

Example 4: Find the least number which leaves a remainder 3 when divided by 4, 5 and 6.
Solution: LCM of 4, 5, 6 = 60. Required number = 60 + 3 = 63.

Example 5: Find the greatest number which divides 245 and 1,029 leaving remainders 5 and 4 respectively.
Solution: Subtract the remainders: 240 and 1025. HCF of 240 and 1025 = 5.

Example 6: Three bells ring at intervals of 12, 15 and 20 minutes. If they ring together now, when will they ring together again?
Solution: LCM of 12, 15, 20 = 60 minutes.

Quick Tips to Solve Faster

• Use HCF x LCM = product of two numbers only for two numbers, not for three.
• For the greatest divisor with remainders, subtract the remainders first and then take the HCF.
• For the least number with the same remainder, use LCM + remainder; for a "short by" case, use LCM - that value.
• If the numbers are in the ratio a : b and the HCF is h, take the numbers as ah and bh.
• For fractions, reduce them to the lowest terms before finding the HCF or LCM.

Common Mistakes to Avoid

• Using HCF x LCM = product of numbers for three numbers.
• Forgetting to subtract the remainders before finding the HCF.
• Taking the highest power for HCF and the lowest for LCM (the opposite is correct).
• Not reducing the fractions to the lowest terms before using the fraction formulas.
• Adding the remainder in the "short by" type questions instead of subtracting.

Quick Revision Points

• HCF x LCM = product of two numbers.
• HCF takes the lowest powers; LCM takes the highest powers.
• Greatest divisor with remainders: HCF of (number - remainder).
• Least number with remainder r: LCM + r.
• HCF of fractions = HCF of numerators / LCM of denominators.

Practice Questions With Answers

1. Find the HCF of 36 and 48.
   Answer: 12 (36 = 2 squared x 3 squared, 48 = 2 to the 4 x 3).
2. Find the LCM of 12, 15 and 20.
   Answer: 60 (2 squared x 3 x 5).
3. The HCF of two numbers is 8, the LCM is 240 and one number is 48. Find the other.
   Answer: 40 (8 x 240 / 48).
4. Find the smallest number divisible by 6, 8 and 12.
   Answer: 24 (LCM).
5. Find the least number which leaves a remainder 5 when divided by 6, 9 and 12.
   Answer: 41 (36 + 5).
6. Find the HCF of 24 and 36.
   Answer: 12 (Common factors).
7. Find the largest number that divides 70 and 105 exactly.
   Answer: 35 (HCF of 70 and 105).
8. Two numbers are in the ratio 3 : 4 and their HCF is 5. Find their LCM.
   Answer: 60 (12 x 5).
9. Three bells ring at intervals of 6, 8 and 10 seconds. After how many seconds will they ring together again?
   Answer: 120 (LCM of 6, 8, 10).
10. Find the smallest 4-digit number divisible by 12.
    Answer: 1,008 (12 x 84).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC fractions and decimals practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Number System Notes
• Ratio and Proportion Notes
• Percentage Notes
• Average Notes
• SSC Fractions and Decimals Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is HCF?

HCF is the Highest Common Factor, the greatest number that divides all the given numbers without a remainder.

What is LCM?

LCM is the Least Common Multiple, the smallest number that is divisible by all the given numbers.

How many questions come from HCF and LCM in SSC exams?

One or two questions usually come directly, and the chapter is also used in other topics.

What is the relation between HCF and LCM of two numbers?

The product of two numbers is equal to the product of their HCF and LCM.

How do I find the HCF of fractions?

HCF of fractions = HCF of the numerators divided by the LCM of the denominators.

How do I solve questions with remainders?

For the greatest divisor, find the HCF of (number - remainder). For the least number, find the LCM and add the common remainder.

What is the HCF of co-prime numbers?

It is always 1.

How do bell and traffic light questions use LCM?

The time after which they act together again is the LCM of their time intervals.

How should I practise HCF and LCM?

Learn the formulas and types, solve the examples here and then practise timed sets of previous year questions.