GCF from Prime Factors
Part of the Factors & Fractions section of Coddy's Pre-Algebra journey. Lesson 13 of 58.
Listing factors gets slow for bigger numbers, and prime factorizations are quicker. Factor both numbers: 45 is 3 × 3 × 5 and 75 is 3 × 5 × 5. The primes they share are one 3 and one 5.
Multiply the shared primes: 3 × 5 is 15, the GCF of 45 and 75. A prime counts as shared only as many times as it appears in both numbers: 45 has two 3s but 75 has only one, so just one 3 goes in.
Why it works: a common factor can only be built from primes both numbers contain. Taking every shared prime, as often as both allow, builds the biggest one. With exponents, take each shared prime with its smaller exponent: 3^2 × 5 and 3 × 5^2 give 3 × 5.
Challenge
Find the greatest common factor g of 84 and 126. Factor both numbers into primes, then write g equal to the product of the primes they share, and work it out on the next line.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Factors & Fractions
1Factors and Multiples
What a Factor IsFactor PairsDivisibility RulesMultiplesPrime or CompositeRecap: Factors and Multiples4What a Fraction Is
Parts of a WholeFractions on the LineA Fraction of a NumberImproper FractionsRecap: What a Fraction Is7Multiplying Fractions
Top Times TopCancel FirstWhole Times FractionA Fraction of a FractionRecap: Multiplying Fractions10Fractions in Stories
Part of a GroupWhat Is LeftSharing a FractionScaling a RecipeRecap: Fraction Stories2Prime Factorization
Factor TreesUsing ExponentsOne Tree per NumberBigger NumbersRecap: Prime Factorization5Equivalent Fractions
Same Amount, New NameSimplifying FractionsSimplify in StepsComparing FractionsRecap: Equivalent Fractions3GCF and LCM
Greatest Common FactorGCF from Prime FactorsLeast Common MultipleLCM from Prime FactorsGCF or LCM?Recap: GCF and LCM