Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2)..
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Factors and multiples return in sixth grade with two specific tools. The greatest common factor (GCF) is the largest whole number that divides both of two numbers, and students find it for numbers up to 100. The least common multiple (LCM) is the smallest number that both numbers divide into, and students find it for numbers up to 12. For 18 and 24, the GCF is 6; for 6 and 8, the LCM is 24.
The second half of the standard links the GCF to the distributive property. A sum such as 36 + 8 can be rewritten as 4(9 + 2), because 4 is a common factor of both numbers. When the factor pulled out is the greatest common factor, the numbers left inside the parentheses share no other factor. This is a small but important step toward algebra: in seventh grade students will factor 24x + 18y as 6(4x + 3y) using the very same idea. In real problems, GCF answers questions about splitting things into the largest equal groups, while LCM answers questions about when repeating events line up again.
Students often give the LCM when asked for the GCF. A quick check helps: the GCF can be no bigger than the smaller number, and the LCM can be no smaller than the larger one.
6 × 8 = 48 is a common multiple, but 24 is the least. Listing multiples of the larger number until one is divisible by the smaller one is reliable.
Writing 36 + 24 as 2(18 + 12) is correct but not fully factored. The GCF, 12, gives 12(3 + 2).
Numbers like 8 and 15 share no factor except 1, so their GCF is 1. Students sometimes say they have no GCF at all.
Find the greatest common factor of 42 and 56, and use it to rewrite 42 + 56.
Answer: The GCF is 14, and 42 + 56 = 14(3 + 4).
Factor rainbows and factor pair tables give students an organized way to list factors without missing any. Venn diagrams of prime factors are a good extension for strong students, but listing is enough for this standard.
Use word problems to make the choice between GCF and LCM explicit: 'largest equal groups' points to the GCF, 'happen together again' points to the LCM. Assessments frequently pair a direct GCF or LCM question with a distributive property rewrite.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 12
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The greatest one in both lists is 12.
Answer: A) 18
Multiples of 9: 9, 18, 27. 18 is the first one that 6 also divides into, so the LCM is 18.
Answer: B) 4(9 + 2)
The GCF of 36 and 8 is 4. 36 = 4 × 9 and 8 = 4 × 2, so 36 + 8 = 4(9 + 2). 2(18 + 4) is correct but does not use the greatest factor.
Answer: 40
This is the LCM of 10 and 8. Multiples of 10: 10, 20, 30, 40. 40 is the first one divisible by 8, so 40 hot dogs (4 packs) and 40 buns (5 packs).
Answer: D) 12
The largest number that divides both 48 and 60 is their GCF, 12. Each kit gets 4 pencils and 5 erasers.
Answer: 1
Factors of 15 are 1, 3, 5, 15 and factors of 28 are 1, 2, 4, 7, 14, 28. The only common factor is 1.
A full lesson with slides, activities and an exit ticket on gcf, lcm and the distributive property, pitched to grade 6 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 6.NS.B.4 with an answer key, ready in about a minute.
Make a worksheet →Turn gcf, lcm and the distributive property into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Greatest common factor problems use whole numbers up to 100, and least common multiple problems use whole numbers up to 12.
Rewriting 36 + 8 as 4(9 + 2) is factoring out a common factor. It builds the skill students use later to factor algebraic expressions.