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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

GCF and LCM from Prime Factorizations

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 21 s
  • British Columbia
  • BC10-MA-06

The video writes 72 = 2³ × 3² and 120 = 2³ × 3 × 5 and sorts their primes into a Venn diagram. The overlap gives the GCF, 2³ × 3 = 24 (the lowest power of each shared prime), and the whole diagram gives the LCM, 2³ × 3² × 5 = 360 (the highest power of every prime). It checks that 24 × 360 = 72 × 120 = 8640, warns against swapping the two rules with a quick size test, and ends on a one-line recap.

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In one sentence

The question this lesson answers

How do we find the GCF and the LCM of 72 and 120?

The GCF, or greatest common factor, is the largest number that is a factor of both.

Transcript

What the lesson says

How do we find the GCF and the LCM of 72 and 120?

The GCF, or greatest common factor, is the largest number that is a factor of both.

The LCM, or least common multiple, is the smallest positive number that is a multiple of both.

A prime factorization writes a number as a product of primes. 72 is 2 times 2 times 2 times 3 times 3.

The exponent, the small raised number, counts the copies: 72 is 2 cubed times 3 squared.

Likewise, 120 is 2 times 2 times 2 times 3 times 5, or 2 cubed times 3 times 5.

In a Venn diagram, the primes both numbers share go in the overlap: 2, 2, 2 and 3.

The other 3 is only in 72, and the 5 is only in 120.

The GCF is the product of the primes in the overlap: 2 times 2 times 2 times 3 is 24.

That is the lowest power of each shared prime: 2 cubed, and 3 rather than 3 squared.

The LCM is the product of every prime in the diagram: 2 times 2 times 2 times 3 times 3 times 5 is 360.

That is the highest power of every prime: 2 cubed, 3 squared and 5.

Check: for two numbers, GCF times LCM equals their product. 24 times 360 and 72 times 120 both make 8640.

Swapping the rules is a common mistake. The GCF is never above the smaller number, nor the LCM below the larger: 24 is under 72, and 360 is over 120.

In short, the GCF takes the lowest powers of shared primes, and the LCM the highest powers of all primes.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Number: prime factorizationBC10-MA-06

Previous: Finding All the Factors of 72 with Factor Pairs

The outcomes and strands follow the British Columbia curriculum.

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