GCF + LCM in Venn Diagrams

Another way we can find greatest common factors (GCFs) and least common multiples (LCMs) is by using a Venn diagram and set theory.

Finding the Greatest Common Factor (GCF)

A prime factorization is basically a set of prime factors multiplied together.

We can find the greatest common factor of number A and number B by comparing their prime factors and finding all of the prime factors they have in common.

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To use a Venn diagram to find the greatest common factor, place the prime factors of in the green circle and the prime factors of in the yellow circle. Place any common prime factors in the middle region where the two circles intersect.

Since the greatest common factor is the set of all common prime factors, the intersection of the two circles is the GCF of and :

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Finding the Least Common Multiple (LCM)

We can find the least common multiple of number A and number B by comparing their prime factors and finding the smallest set of prime factors that contain all of the prime factors in number A and all of the prime factors in number B.

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To use a Venn diagram to find the least common multiple, place the prime factors of in the green circle and the prime factors of in the yellow circle. Place any common prime factors in the middle region where the two circles intersect.

Since the least common multiple is the smallest set of prime factors that contains all of the prime factors in the green circle and all of the prime factors in the yellow circle, the union of the two circles is the LCM of and :

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