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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 5-6, Math Kangaroo 7-8.

LCM - Least Common Multiple of a Set of Natural Numbers

Given a set of natural numbers, the least common multiple is computed by:

- computing all the prime factorizations
- multiplying together all the distinct prime factors that exist in the prime factorizations
- raising each prime to the largest power it appears at in any of the prime factorizations

Example:

The least common multiple is the smallest integer that divides exactly into all the numbers present in the set.

The least common multiple is used primarily to determine the common denominator of a set of fractions.

Example: An architect has been asked to design a high-rise building that can accommodate either 14 suites per story or 12 suites per story. What is the minimum number of stories the building could have?

The overall number of suites has to divisible by both 12 and 14. It must therefore be a multiple of both numbers. The smallest such multiple is the LCM:

The smallest number of stories that accommodates 84 suites is 6:

GCF - Greatest Common Factor of a Set of Natural Numbers

Given a set of natural numbers, the greatest common factor is computed by:

- computing all the prime factorizations
- multiplying together all the prime factors that occur in all the prime factorizations
- raising each prime to the smallest power it appears at in any of the prime factorizations

Example:

The greatest common factor is the largest number that divides exactly all the numbers present in the set.

The GCF is mainly used in order to determine how to simplify fractions to render them irreducible.