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Quantitative AptitudeBeginner4 min study

How to Find the HCF of Three Numbers: Step-by-Step Guide

तीन संख्याओं का म.स. (HCF) कैसे निकालें: आसान तरीका

The Highest Common Factor (HCF), or महत्तम समापवर्तक (म.स.), also known as the Greatest Common Divisor (GCD), of three numbers is the largest positive number that divides all three numbers completely with zero remainder. In competitive exams, HCF is fundamental for simplifying fractions, finding maximum tile sizes, and grouping problems.

Key Rules & Core Formulas

Understand these foundational properties before solving questions

1Definition of HCF

HCF(a, b, c) = Greatest number that divides a, b, and c

The HCF can never be greater than the smallest of the three numbers.

If three numbers share no common factor other than 1, their HCF is 1 (they are co-prime).

2Prime Factorization Method for HCF

HCF = Product of lowest powers of common prime factors only

Only include prime factors that appear in ALL three factorizations, raised to their lowest power.

Unlike LCM which takes highest powers of all factors, HCF takes lowest powers of common factors only.
Standard Method

Step-by-Step Method to Solve

Follow these universal steps in order to solve any Quantitative Aptitude question accurately:

  1. 1

    Step 1: Find the prime factors of each number

    Break down each of the three numbers into their prime factor representations.

    Pro-Tip: Write powers clearly (e.g., 2³ instead of 2 × 2 × 2) to easily spot the lowest power.
  2. 2

    Step 2: Identify prime factors common to all three numbers

    Circle only those prime numbers that appear in the factorization of all three numbers. If a prime factor is missing in even one number, discard it.

    Pro-Tip: If no prime number appears in all three factorizations, the HCF is 1.
  3. 3

    Step 3: Select the lowest power of each common prime factor

    For each common prime factor identified, take the smallest exponent among the three representations.

    Pro-Tip: Lowest power ensures the factor divides all three numbers.
  4. 4

    Step 4: Multiply the lowest powers together

    Multiply the selected lowest powers to get the final HCF value.

    Pro-Tip: Check that the final HCF divides each of the original three numbers without remainder.

Worked Examples & Walkthroughs

Real questions solved step-by-step showing every calculation

1Example 1Method: Prime Factorization Method

Question

Find the HCF of 24, 36, and 60.

Step-by-Step Solution:

Step 1: Factorize each number:24 = 2³ × 3¹, 36 = 2² × 3², 60 = 2² × 3¹ × 5¹.
Step 2: Find common primes:Primes common to all three numbers are 2 and 3. (5 is only in 60, so discard it).
Step 3: Take lowest powers:Lowest power of 2 is 2². Lowest power of 3 is 3¹.
Step 4: Multiply:HCF = 2² × 3¹ = 4 × 3 = 12.
Final Result:
HCF(24, 36, 60) = 12
Exam Shortcut / Verification: Verification: 24 ÷ 12 = 2, 36 ÷ 12 = 3, 60 ÷ 12 = 5. All quotient numbers (2, 3, 5) share no common factor.

Common Mistakes to Avoid

Avoid these frequent traps to protect yourself from negative marking

1

Including prime factors that are not present in all three numbers

Why it happens: Confusing HCF rules with LCM rules.
Correct approach: For HCF, a prime factor MUST appear in all three numbers. If a number lacks it, that factor cannot be in the HCF.
2

Selecting the highest power instead of the lowest power

Why it happens: The word 'Highest' in HCF confuses students into picking highest powers.
Correct approach: Remember the inverted rule: Highest Common Factor takes the LOWEST power!
Self-Assessment

Try It Yourself (Practice Questions)

Test your understanding on your own first, then reveal the step-by-step solution to check your work:

Practice #1

Find the HCF of 18, 45, and 63.

Quick Revision Summary

  • HCF is the largest factor dividing all three numbers.
  • Factorize all three numbers into prime factors.
  • Pick only common prime factors raised to their lowest power.
  • Multiply them to find the HCF.
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