How to Find a Missing Number Using HCF and LCM

What formula should you use?

For two positive integers a and b, HCF(a,b) × LCM(a,b) = ab. If one number is known, divide the product of the HCF and LCM by that number to find a candidate for the other. Then check the actual HCF and LCM, because inconsistent data can still produce an integer candidate.

Example: HCF 6, LCM 180, one number 30

The other number is (6 × 180)/30 = 36. Check using prime factors: 30 = 2 × 3 × 5 and 36 = 2² × 3². The common lowest powers give HCF = 2 × 3 = 6. The highest powers give LCM = 2² × 3² × 5 = 180. Both conditions hold.

When the candidate does not work

Suppose the HCF is 6, the LCM is 72 and one number is 18. The formula gives the other as 24. Their actual HCF is 6 and LCM is 72, so this is valid. But with HCF 6, LCM 90 and one number 18, the candidate is 30: its actual HCF is 6 and LCM is 90, also valid.

Now consider HCF 6, LCM 36 and one number 12. The candidate is 18, which works. By contrast, HCF 6, LCM 24 and one number 12 gives candidate 12. Two identical numbers 12 and 12 have HCF 12 and LCM 12, not the stated values. The product equation is necessary but is not a complete consistency test.

Important limits

This simple identity applies to two positive integers. Do not extend HCF × LCM = product directly to three numbers. The HCF must divide both numbers, and each number must divide the LCM. These are useful early checks, but the final pair should still be verified.

Practice

1. HCF 4, LCM 60, one number 12: 20.
2. HCF 5, LCM 100, one number 20: 25.
3. HCF 4, LCM 16, one number 8: candidate 8 is invalid because HCF(8,8) = 8.

Report inconsistent information when necessary instead of accepting every calculated quotient.

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