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HCF and LCM

Practice MCQs

HCF and LCM are useful in various mathematical computations, such as simplifying fractions, solving word problems, and finding common denominators. They are fundamental concepts in number theory and play a significant role in various mathematical operations and applications.

Quantitative Aptitude HCF and LCM Competitive Exams

HCF and LCM are important number system concepts used in quantitative aptitude. HCF helps us find the greatest common factor of numbers, while LCM helps us find the least common multiple. These concepts are frequently used in simplification, fractions, time intervals, bells, divisibility, and word problems.

What are HCF and LCM?

HCF stands for Highest Common Factor. It is the greatest number that divides two or more given numbers exactly.

LCM stands for Least Common Multiple. It is the smallest number that is exactly divisible by two or more given numbers.

Quick idea: HCF is used when we need the greatest possible common divisor. LCM is used when we need the smallest common multiple or repeated meeting point.
Concept Meaning Example
Factor A number that divides another number exactly Factors of 12: 1, 2, 3, 4, 6, 12
Multiple A number obtained by multiplying a given number Multiples of 6: 6, 12, 18, 24
HCF Greatest common factor HCF of 12 and 18 is 6
LCM Smallest common multiple LCM of 12 and 18 is 36

“HCF divides the numbers; LCM is divisible by the numbers.”

Aptitude Tip
Key points
  • HCF is the greatest common divisor.
  • LCM is the smallest common multiple.
  • HCF is always less than or equal to the smallest number.
  • LCM is always greater than or equal to the largest number.
  • Prime factorization is useful for both HCF and LCM.
  • For two numbers, product of numbers = HCF × LCM.
factors multiples divisibility prime factors

Visual Understanding

These diagrams show how HCF and LCM are understood using factors, multiples, and prime factorization.

HCF: Common Factors
Factors of 12 Factors of 18 4 12 1 2 3 6 9 18 Greatest common factor = 6
\[ HCF(12,18)=6 \]

HCF is the greatest number present in the common factor region.

LCM: Common Multiples
6 12 18 24 30 36 First common multiple Example: LCM of 6 and 8 is 24
\[ LCM(6,8)=24 \]

LCM is the smallest number that both given numbers divide exactly.

Prime Factor Tree
36 6 6 2 3 2 3
\[ 36 = 2^2 \times 3^2 \]

Prime factorization is one of the best methods to find HCF and LCM.

Important Formulas and Rules

Product Rule
\[ a \times b = HCF(a,b) \times LCM(a,b) \]

This rule works for two numbers.

LCM from HCF
\[ LCM = \frac{a \times b}{HCF} \]

Useful when two numbers and HCF are known.

HCF from LCM
\[ HCF = \frac{a \times b}{LCM} \]

Useful when two numbers and LCM are known.

Co-prime Numbers
\[ HCF(a,b)=1 \]

If two numbers have no common factor except 1, they are co-prime.

Rule: In prime factorization, HCF uses the common prime factors with the smallest powers, while LCM uses all prime factors with the greatest powers.

Methods to Find HCF and LCM

Listing Method

List factors or multiples and identify the common value.

  • Easy for small numbers
  • List factors for HCF
  • List multiples for LCM
  • Time-consuming for large numbers
Prime Factorization

Break each number into prime factors.

  • Best for medium numbers
  • Compare prime powers
  • Small powers for HCF
  • Large powers for LCM
Division Method

Use repeated division by common prime factors.

  • Useful for multiple numbers
  • Fast for LCM
  • Common divisors help HCF
  • Popular in exams
Euclidean Method

Repeated division method for HCF.

  • Useful for large numbers
  • Divide larger by smaller
  • Continue with remainder
  • Last divisor is HCF
Exam approach: For small numbers, listing method is fine. For larger numbers, prime factorization or division method is faster.
Prime Factorization Method

Suppose we need to find HCF and LCM of \(24\) and \(36\).

\[ 24 = 2^3 \times 3 \]
\[ 36 = 2^2 \times 3^2 \]

For HCF, take common prime factors with smaller powers:

\[ HCF = 2^2 \times 3 = 12 \]

For LCM, take all prime factors with greater powers:

\[ LCM = 2^3 \times 3^2 = 72 \]

Tip: HCF chooses minimum powers; LCM chooses maximum powers.

Division Method Example

Find HCF of \(48\) and \(18\) using Euclidean method.

Step Division Remainder
1 \(48 \div 18\) \(12\)
2 \(18 \div 12\) \(6\)
3 \(12 \div 6\) \(0\)
\[ HCF(48,18)=6 \]

When the remainder becomes zero, the last divisor is the HCF.

Step-by-Step Solving Method

Step For HCF For LCM
Step 1 Write prime factorization of each number. Write prime factorization of each number.
Step 2 Choose only common prime factors. Choose all prime factors.
Step 3 Take the smallest powers of common factors. Take the greatest powers of all factors.
Step 4 Multiply selected factors. Multiply selected factors.
Step 5 Answer is the greatest common factor. Answer is the least common multiple.
Important: HCF is connected with division and sharing. LCM is connected with repetition, timing, and common occurrence.

Solved Examples

Question Method Answer
Find the HCF of 12 and 18. Factors of 12: \(1,2,3,4,6,12\)
Factors of 18: \(1,2,3,6,9,18\)
Common factors: \(1,2,3,6\) 		6
Find the LCM of 12 and 18.
\[ 12 = 2^2 \times 3 \]
\[ 18 = 2 \times 3^2 \]
\[ LCM = 2^2 \times 3^2 = 36 \]
36
Find the HCF and LCM of 24 and 36.
\[ 24 = 2^3 \times 3 \]
\[ 36 = 2^2 \times 3^2 \]
\[ HCF = 2^2 \times 3 = 12 \]
\[ LCM = 2^3 \times 3^2 = 72 \]
HCF = 12, LCM = 72
The HCF of two numbers is 6 and their LCM is 72. If one number is 24, find the other number.
\[ a \times b = HCF \times LCM \]
\[ 24 \times b = 6 \times 72 \]
\[ b = \frac{432}{24} = 18 \]
18
Find the HCF of 48 and 18 using division method. \(48 = 18 \times 2 + 12\)
\(18 = 12 \times 1 + 6\)
\(12 = 6 \times 2 + 0\)
Last divisor is \(6\). 		6
Find the LCM of 8, 12, and 20.
\[ 8 = 2^3 \]
\[ 12 = 2^2 \times 3 \]
\[ 20 = 2^2 \times 5 \]
\[ LCM = 2^3 \times 3 \times 5 = 120 \]
120
Find the HCF of 45, 60, and 75.
\[ 45 = 3^2 \times 5 \]
\[ 60 = 2^2 \times 3 \times 5 \]
\[ 75 = 3 \times 5^2 \]
\[ HCF = 3 \times 5 = 15 \]
15
Two bells ring at intervals of 12 minutes and 18 minutes. After how many minutes will they ring together? This is a repeated event problem, so use LCM.
\[ LCM(12,18)=36 \]
36 minutes

Note: Use HCF for maximum equal grouping. Use LCM for repeated events or first common occurrence.

Common Traps and Shortcuts

Common Traps
  • Confusing factors with multiples.
  • Using LCM when the question asks for greatest possible size.
  • Using HCF when the question asks for repeated meeting time.
  • Forgetting that HCF uses common factors only.
  • Taking smallest powers for LCM by mistake.
  • Applying product rule to more than two numbers without care.
Useful Shortcuts
  • HCF is usually used for dividing into equal groups.
  • LCM is usually used for repeated events.
  • For HCF, take common prime factors with smallest powers.
  • For LCM, take all prime factors with greatest powers.
  • For two numbers, use \(a \times b = HCF \times LCM\).
  • For bells, traffic lights, and meeting intervals, use LCM.
Exam approach: If the question asks for greatest possible equal size, use HCF. If it asks for the first common time or repetition, use LCM.

Practice

A) Multiple Choice Questions
  1. Find the HCF of 16 and 24.
    4 6 8 12
  2. Find the LCM of 6 and 8.
    12 18 24 48
  3. Find the HCF of 45 and 60.
    5 10 15 20
  4. Find the LCM of 10, 15, and 20.
    30 40 60 120
  5. If HCF of two numbers is 4 and LCM is 60, their product is:
    120 180 240 300
B) Solve the Higher-Order Problems
  1. Find the HCF and LCM of 18 and 30. (Hint: Use prime factorization.)
  2. Three bells ring at intervals of 6 minutes, 8 minutes, and 12 minutes. After how many minutes will they ring together? (Hint: Use LCM.)
  3. The HCF of two numbers is 5 and their LCM is 150. If one number is 25, find the other number. (Hint: \(a \times b = HCF \times LCM\).)
  4. Find the greatest number that divides 48, 72, and 96 exactly. (Hint: Greatest divisor means HCF.)
  5. Find the least number that is divisible by 9, 12, and 15. (Hint: Least divisible number means LCM.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
HCF Greatest common factor
LCM Least common multiple
Factor Number that divides exactly
Multiple Number obtained by repeated multiplication
Co-prime numbers Numbers having HCF equal to 1
Bells ringing together LCM-based problem
Aptitude Reminder

HCF is used when the problem asks for the greatest possible equal division. LCM is used when the problem asks for the least common occurrence or repetition. Prime factorization helps solve most HCF and LCM questions quickly.

Task: Create five questions using factor listing, prime factorization, product rule, bells interval, and greatest possible divisor.

Show Suggested Answers
Multiple Choice
  1. 8
    Factors of 16: \(1,2,4,8,16\).
    Factors of 24: \(1,2,3,4,6,8,12,24\).
    Greatest common factor = 8.
  2. 24
    Multiples of 6: \(6,12,18,24\).
    Multiples of 8: \(8,16,24\).
    Least common multiple = 24.
  3. 15
    \[ 45 = 3^2 \times 5 \]
    \[ 60 = 2^2 \times 3 \times 5 \]
    \[ HCF = 3 \times 5 = 15 \]
  4. 60
    \[ 10 = 2 \times 5 \]
    \[ 15 = 3 \times 5 \]
    \[ 20 = 2^2 \times 5 \]
    \[ LCM = 2^2 \times 3 \times 5 = 60 \]
  5. 240
    \[ \text{Product} = HCF \times LCM = 4 \times 60 = 240 \]
Higher-Order Problems
  1. Find HCF and LCM of 18 and 30:
    \[ 18 = 2 \times 3^2 \]
    \[ 30 = 2 \times 3 \times 5 \]
    \[ HCF = 2 \times 3 = 6 \]
    \[ LCM = 2 \times 3^2 \times 5 = 90 \]
    Answer = HCF = 6, LCM = 90.
  2. Bells ring together after LCM of \(6,8,12\):
    \[ 6 = 2 \times 3,\quad 8 = 2^3,\quad 12 = 2^2 \times 3 \]
    \[ LCM = 2^3 \times 3 = 24 \]
    Answer = 24 minutes.
  3. HCF = 5, LCM = 150, one number = 25:
    \[ 25 \times b = 5 \times 150 \]
    \[ b = \frac{750}{25} = 30 \]
    Answer = 30.
  4. Greatest number dividing \(48,72,96\) exactly means HCF:
    \[ HCF(48,72,96)=24 \]
    Answer = 24.
  5. Least number divisible by \(9,12,15\) means LCM:
    \[ 9 = 3^2,\quad 12 = 2^2 \times 3,\quad 15 = 3 \times 5 \]
    \[ LCM = 2^2 \times 3^2 \times 5 = 180 \]
    Answer = 180.
Concept Matching
  1. HCF → Greatest common factor
  2. LCM → Least common multiple
  3. Factor → Number that divides exactly
  4. Multiple → Number obtained by repeated multiplication
  5. Co-prime numbers → Numbers having HCF equal to 1
  6. Bells ringing together → LCM-based problem
Clue Explanation

HCF is used for greatest equal division. LCM is used for smallest common repetition. This distinction helps identify the correct method in word problems.

Exam tips
  • Use HCF for maximum equal grouping.
  • Use LCM for repeated events and common timing.
  • For HCF, use common prime factors only.
  • For LCM, use all prime factors.
  • Remember: smaller powers for HCF, greater powers for LCM.
  • For two numbers, product = HCF × LCM.
Practice MCQs