HCF & LCM Applications | Real Numbers | Class 10 Mathematics

01 · KEY FORMULA

Relationship Between HCF, LCM & Two Numbers

For any two positive integers a and b, the product of their HCF and LCM is equal to the product of the numbers:

HCF(a, b) × LCM(a, b) = a × bNote: This property holds true ONLY for two numbers (not for three or more numbers!).

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Rearranged Formulas

1. LCM(a, b) = (a × b) / HCF(a, b)
2. HCF(a, b) = (a × b) / LCM(a, b)

02 · SOLVED NCERT EXAMPLES

Step-by-Step Word Problems

EXAMPLE 1

Given HCF(306, 657) = 9, find LCM(306, 657)

Given: a = 306, b = 657, HCF(a,b) = 9.

Using the property: LCM(a, b) = (a × b) / HCF(a, b)

LCM(306, 657) = (306 × 657) / 9
= 34 × 657 = 22338

EXAMPLE 2 (NCERT WORD PROBLEM)

Circular Track Problem

Problem: Sonia takes 18 minutes to drive one lap, while Ravi takes 12 minutes for the same. If they start at the same point and time in the same direction, after how many minutes will they meet again at the starting point?

Solution: The time after which they meet again is the LCM of 18 and 12.

18 = 2¹ × 3²
12 = 2² × 3¹
LCM(18, 12) = 2² × 3² = 4 × 9 = 36

Conclusion: They will meet again at the starting point after 36 minutes.

03 · QUICK SUMMARY

Key Takeaways

HCF(a,b) × LCM(a,b) = a × b holds for any two positive integers.

For meeting at starting point / repeating intervals, calculate LCM.

For maximum stack size / equal grouping without leftovers, calculate HCF.