LCM and HCF for CUET UG GAT 2026 – Formulas, Tricks, Questions, and Solutions

Introduction to LCM and HCF for CUET UG GAT 2026

LCM and HCF for CUET UG GAT 2026 – Formulas, Tricks, Questions, and Solutions: LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental topics in the Number System section of the CUET UG GAT 2026 syllabus. These concepts are widely used in solving problems related to fractions, ratios, time intervals, and divisibility.

In CUET exams, 1–2 questions are regularly asked from LCM and HCF. Students who understand the formulas and shortcuts can solve these questions quickly and accurately.

This guide covers:

• Concepts and formulas
• Short tricks
• Common patterns
• 25 original MCQs with step-by-step solutions
• FAQs for quick revision

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What is LCM and HCF?

HCF (Highest Common Factor)

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides all numbers exactly.

Example

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

HCF = 6

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LCM (Least Common Multiple)

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is divisible by all numbers.

Example

Multiples of 4:
4, 8, 12, 16, 20

Multiples of 6:
6, 12, 18, 24

Common multiple:
12

LCM = 12

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Important Formulas for LCM and HCF

(1) Relationship Formula

LCM × HCF = Product of Numbers

Example:

Numbers = 12 and 15

HCF = 3

LCM = (12 × 15) / 3

LCM = 60

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(2) HCF by Prime Factorization

Steps:

1. Find prime factors
2. Take common factors
3. Multiply them

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(3) LCM by Prime Factorization

Steps:

1. Find prime factors
2. Take highest powers
3. Multiply them

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Short Tricks to Solve LCM and HCF Questions

Trick 1

If numbers are co-prime

HCF = 1

LCM = Product of numbers

Example:

8 and 15

LCM = 8 × 15

LCM = 120

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Trick 2

To find HCF quickly:

Divide numbers repeatedly using common prime numbers.

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Trick 3

For fractions:

LCM of fractions =

LCM of numerators / HCF of denominators

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Trick 4

If one number divides another:

Smaller number = HCF
Larger number = LCM

Example:

5 and 20

HCF = 5
LCM = 20

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Common Patterns of LCM and HCF Questions in CUET UG GAT

1. Find LCM or HCF of numbers
2. Find smallest number divisible by given numbers
3. Find greatest number dividing numbers
4. Word problems on time intervals
5. Fractions and ratios
6. Missing number problems
7. Product relationship questions

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11 Important Exam-Oriented LCM and HCF MCQs for CUET UG GAT 2026

Q 4471) Find the LCM of 9, 12, and 15.

(A) 120
(B) 130
(C) 160
(D) 180

Answer

(D) 180

Solution

9 = 3²
12 = 2² × 3
15 = 3 × 5

LCM:

2² × 3² × 5

LCM = 180

Q 4472) Find the HCF of 72 and 120.

(A) 12
(B) 18
(C) 24
(D) 36

Answer

(C) 24

Solution

72 = 2³ × 3²
120 = 2³ × 3 × 5

Common factors:

2³ × 3

HCF = 24

Q 4473) Find the greatest number that divides 45, 60, and 75 exactly.

(A) 15
(B) 18
(C) 20
(D) 25

Answer

(A) 15

Solution

45 = 3² × 5
60 = 2² × 3 × 5
75 = 3 × 5²

Common factor:

3 × 5

HCF = 15

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Q 4474) The LCM of two numbers is 240 and their HCF is 12. Find the product of the numbers.

(A) 2480
(B) 2650
(C) 2880
(D) 3050

Answer

(C) 2880

Solution

Product:

LCM × HCF

= 240 × 12

= 2880

Q 4475) The HCF of two numbers is 9 and their LCM is 180. If one number is 45, find the other number.

(A) 36
(B) 48
(C) 60
(D) 72

Answer

(A) 36

Solution

Formula:

Product of numbers = HCF × LCM

= 9 × 180

= 1620

Other number:

1620 ÷ 45

= 36

Q 4476) Find the least number which when divided by 5, 6, and 8 leaves remainder 3 in each case.

(A) 121
(B) 123
(C) 117
(D) 115

Answer

(B) 123

Solution

LCM of:

5, 6, 8

= 120

Required number:

120 + 3

= 123

Q 4477) Find the greatest number that divides 81 and 153 leaving the same remainder.

(A)  9
(B) 12
(C) 15
(D) 18

Answer

(A) 9

Solution

Difference:

153 − 81 = 72

HCF of:

81 and 72

= 9

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Q 4478) Find the least number which when divided by 10, 15, and 25 leaves remainder 2 in each case.

(A) 145
(B) 148
(C) 152
(D) 156

Answer

(C) 152

Solution

LCM of:

10, 15, 25

= 150

Required number:

150 + 2

= 152

Q 4479) Find the smallest number divisible by 6, 8, and 12.

(A) 18
(B) 20
(C) 22
(D) 24

Answer

(D) 24

Solution

6 = 2 × 3
8 = 2³
12 = 2² × 3

LCM:

2³ × 3

LCM = 24

Q 4480) Find the HCF of 24 and 36.

(A) 10
(B) 12
(C) 16
(D) 18

Answer

(B) 12

Solution

24 = 2³ × 3
36 = 2² × 3²

Common factors:

2² × 3

HCF = 12

Q 4481) The LCM of two numbers is 84 and their HCF is 7. Find the product of the numbers.

(A) 488
(B) 496
(C) 556
(D) 588

Answer

(D) 588

Solution

Product:

LCM × HCF

= 84 × 7

= 588

Why LCM and HCF Topic is Important for CUET UG GAT 2026

1. Frequently asked in CUET exams
2. Builds strong Number System foundation
3. Helps solve Time and Work problems
4. Used in Fractions and Ratios
5. Easy scoring topic
6. Improves calculation speed
7. Important for competitive exams

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FAQs on LCM and HCF for CUET UG GAT

(1) What is the formula of LCM and HCF?

LCM × HCF = Product of Numbers

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(2) Is LCM and HCF important for CUET UG GAT?

Yes. It is a core topic in the Number System section and regularly appears in the exam.

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(3) How many questions come from this topic in CUET UG GAT?

Usually:

1–2 questions

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(4) Which method is fastest to find HCF?

Prime factorization or division method.

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(5) What is the HCF of co-prime numbers?

HCF = 1

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Conclusion

LCM and HCF are essential topics for CUET UG GAT 2026 preparation. By understanding formulas, shortcuts, and patterns, students can solve questions quickly and score higher marks. Regular practice of MCQs and application of tricks will significantly improve accuracy and speed in the exam.

Mastering this topic strengthens your overall Number System concepts and builds confidence for solving advanced quantitative aptitude problems.

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