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H.C.F. & L.C.M.

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Quantitative Aptitude - H.C.F. & L.C.M.

Understand, apply, and solve quickly to score easy marks.

1. What are H.C.F. and L.C.M.?

  • H.C.F. (Highest Common Factor): Greatest number that divides all given numbers exactly.
  • L.C.M. (Least Common Multiple): Smallest number that is divisible by all given numbers.

2. Relationship Between H.C.F. and L.C.M.:

\[ \text{H.C.F.}(a,b) \times \text{L.C.M.}(a,b) = a \times b \]

3. Methods to Calculate:

  • Prime Factorization: Express numbers as products of primes; take minimum powers for H.C.F., maximum for L.C.M.
  • Division/Ladder Method: Divide both numbers by common primes step by step until no common divisors left.

4. How to Solve Word Problems:

  • If objects/events repeat after certain cycles, use L.C.M. to find the repetition interval.
  • If multiple groups divide things equally, use H.C.F. to determine maximum equal group size.

5. Tips to Answer Quickly:

  • Start with H.C.F.; then use the formula to find L.C.M. to save time.
  • For more than two numbers, combine stepwise: H.C.F.(a,b,c) = H.C.F.(H.C.F.(a,b), c).
  • Use divisibility shortcuts (for 2,3,5,7) to speed up prime factorization.
  • Keep factor tables ready for small primes (2,3,5,7,11).

6. Typical Exam Questions:

  • Find H.C.F. and L.C.M. of multiple numbers.
  • Find least value divisible by given numbers.
  • Find maximum size or count in division problems.

7. Revision Checklist:

  • Know the identity: H.C.F. × L.C.M. = product of numbers.
  • Be fluent in prime factorization up to at least 100.
  • Practice at least 15 H.C.F./L.C.M. problems every week.

Summary: Break numbers into primes, identify common and uncommon powers, apply the identity, and use H.C.F./L.C.M. logic in word problems for fast and accurate solutions.

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