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Quantitative AptitudeBeginner3 min study

How to Use the Relation Between LCM and HCF in Exam Questions

LCM और HCF के बीच संबंध: परीक्षा में प्रश्न हल करने का तरीका

One of the most frequently asked question patterns in state competitive exams is based on the golden mathematical relation between the LCM and HCF of two numbers. Knowing how to manipulate this formula saves valuable minutes during exams.

Key Rules & Core Formulas

Understand these foundational properties before solving questions

1Golden Formula (Two Numbers)

First Number (a) × Second Number (b) = LCM(a, b) × HCF(a, b)

The product of any two positive integers is always equal to the product of their LCM and HCF.

This identity is strictly valid for TWO numbers only. It does not apply directly to three numbers.

2Derived Formula for One Unknown Number

Other Number = (LCM × HCF) / Given Number

If LCM, HCF, and one of the two numbers are given, you can find the missing number directly using simple division.

Always check if the given number divides the product of LCM and HCF without remainder.
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: Identify the given values from the question

    Read the problem carefully and label the known quantities: HCF, LCM, and the known number (a).

    Pro-Tip: Write them down: HCF = ..., LCM = ..., a = ..., b = ?
  2. 2

    Step 2: Set up the formula

    Substitute the values into the formula: a × b = LCM × HCF.

    Pro-Tip: Rearrange to: b = (LCM × HCF) / a.
  3. 3

    Step 3: Simplify by canceling before multiplying

    Do NOT multiply large numbers first! Cancel the denominator with either the LCM or HCF to keep calculations fast and error-free.

    Pro-Tip: Canceling first avoids large multiplication and prevents arithmetic errors.
  4. 4

    Step 4: Calculate the final answer and verify

    Multiply the remaining factors to get the second number.

    Pro-Tip: Verify: HCF must divide both numbers evenly.

Worked Examples & Walkthroughs

Real questions solved step-by-step showing every calculation

1Example 1Method: Product Formula Method

Question

The HCF of two numbers is 12, and their LCM is 144. If one of the numbers is 36, find the other number.

Step-by-Step Solution:

Step 1: Identify values:HCF = 12, LCM = 144, First Number (a) = 36, Second Number (b) = ?
Step 2: Apply formula:b = (LCM × HCF) / a = (144 × 12) / 36.
Step 3: Simplify by canceling:Divide 36 by 12: 36 ÷ 12 = 3. Now the equation becomes: b = 144 / 3.
Step 4: Compute final value:144 ÷ 3 = 48.
Final Result:
The other number is 48
Exam Shortcut / Verification: Notice 36 is 3 × 12. Since 144 ÷ 3 = 48, the answer is 48 in just 5 seconds!

Common Mistakes to Avoid

Avoid these frequent traps to protect yourself from negative marking

1

Applying the formula (a × b × c = LCM × HCF) to three numbers

Why it happens: Assuming the two-number formula holds true for three or more numbers.
Correct approach: Never apply a × b × c = LCM × HCF for three numbers! The product of three numbers is NOT equal to LCM × HCF.
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

The LCM of two numbers is 864 and their HCF is 144. If one number is 288, what is the other number?

Quick Revision Summary

  • Product of two numbers = LCM × HCF.
  • Missing number = (LCM × HCF) / Given Number.
  • Always cancel before multiplying to save time.
  • This formula works ONLY for two numbers.
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