How to Solve HCF and LCM Word Problems
Solve HCF and LCM word problems using keyword recognition, prime factorization, and the HCF×LCM=product identity, with a worked example.
Expected Interview Answer
HCF (highest common factor) finds the largest quantity that divides all given numbers evenly, used for 'splitting into identical maximum-size groups’ problems, while LCM (least common multiple) finds the smallest quantity divisible by all given numbers, used for 'events realigning' or 'minimum common quantity' problems.
HCF problems typically involve phrases like 'largest possible size,' 'maximum number of equal groups,' or 'greatest length that measures exactly' — the answer is the HCF of the given quantities. LCM problems involve phrases like 'smallest time when events coincide again,' 'least number divisible by all,' or 'minimum quantity that can be split evenly by each' — the answer is the LCM. A key identity ties them together for two numbers: HCF × LCM = product of the two numbers, which lets you find one from the other three quantities. For three or more numbers, this identity does not directly apply, and both HCF and LCM must be computed via prime factorization or the Euclidean algorithm.
- Keyword recognition (largest/greatest → HCF, smallest/least → LCM) picks the right tool instantly
- HCF × LCM = product shortcut solves missing-value problems for two numbers
- Prime factorization method generalizes cleanly to three or more numbers
AI Mentor Explanation
Cutting a training ground into the largest possible identical square practice zones, when the ground is 48m by 36m, requires finding the greatest square side that divides both dimensions evenly — that greatest side is the HCF of 48 and 36, which is 12m. If two bowlers’ run-up cycles are 8 and 12 balls respectively and you want to know when both start their run-up together again, that is the LCM of 8 and 12, giving 24 balls. HCF answers 'largest exact-fit division'; LCM answers 'smallest point where cycles realign.'
Worked example
Prime factorize
- 6=2×3
- 8=2³
- 12=2²×3
Take highest powers
- LCM = 2³×3 = 24
Answer
- Next together at 9:24
Step-by-Step Explanation
Step 1
Identify the keyword
"Largest/greatest/maximum equal division" → HCF; "smallest/least/simultaneous recurrence" → LCM.
Step 2
Prime factorize each number
Break each given number into its prime factors.
Step 3
Compute HCF or LCM
HCF = product of lowest powers of common primes; LCM = product of highest powers of all primes present.
Step 4
Apply the identity for two numbers
HCF × LCM = product of the two numbers, useful to find a missing value.
What Interviewer Expects
- Correct keyword-to-operation mapping (largest→HCF, smallest→LCM)
- Accurate prime factorization and highest/lowest power selection
- Correct use of the HCF × LCM = product identity for two-number problems
- Awareness that the identity does not extend directly to three or more numbers
Common Mistakes
- Confusing HCF and LCM keywords, e.g. using LCM for a “largest equal group” problem
- Applying HCF × LCM = product to three or more numbers incorrectly
- Taking the wrong power (highest instead of lowest, or vice versa) during factorization
- Forgetting to add the starting offset when a problem asks for the next simultaneous occurrence
Best Answer (HR Friendly)
“The trick is spotting the keyword. If the problem wants the largest possible equal split or the biggest measuring unit, that is HCF. If it wants the earliest point where repeating things line up again, that is LCM. Once I know which one, I prime-factorize the numbers, and for HCF I take the lowest shared powers, for LCM I take the highest powers across all of them. For two numbers, I can also cross-check using HCF times LCM equals the product of the numbers.”
Follow-up Questions
- How do you find the HCF and LCM of more than two numbers?
- How does the Euclidean algorithm compute HCF faster than prime factorization for large numbers?
- Can you have HCF × LCM = product for three numbers? Why or why not?
- How would you find the smallest number that leaves a specific remainder when divided by several given numbers?
MCQ Practice
1. Find the HCF of 36 and 60.
36=2²×3², 60=2²×3×5. Common lowest powers: 2²×3 = 12.
2. Two numbers have HCF 4 and LCM 84. If one number is 12, the other is?
HCF × LCM = product, so 4×84 = 12 × other → other = 336/12 = 28.
3. Find the smallest number exactly divisible by 4, 6, and 9.
4=2², 6=2×3, 9=3². LCM = 2²×3² = 36.
Flash Cards
Keyword for HCF problems? — Largest / greatest / maximum equal division.
Keyword for LCM problems? — Smallest / least / earliest simultaneous recurrence.
HCF × LCM identity? — For two numbers: HCF × LCM = product of the two numbers.
HCF via prime factorization? — Product of the lowest common powers of shared primes.