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Greatest Common Factor & Least Common Multiple cheat sheet - grade 5-7

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Math Grade 5-7

Greatest Common Factor & Least Common Multiple Cheat Sheet

A printable reference covering prime factorization, greatest common factor, least common multiple, factor pairs, and the relationship $a \times b = \mathrm{GCF} \times \mathrm{LCM}$ for grades 5-7.

TopicsAlgebraMultiplicationDivisionExponentsStudy GuideMathematics
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Greatest common factor and least common multiple help students compare whole numbers, simplify fractions, and solve real-world grouping and scheduling problems. This cheat sheet covers prime factorization, listing factors and multiples, and choosing whether a problem needs GCF or LCM. It gives students a quick way to organize numbers so they can avoid guessing.

These skills are especially useful before learning more advanced fraction and ratio work.

The greatest common factor, or GCF, is the largest factor shared by two or more numbers. The least common multiple, or LCM, is the smallest positive multiple shared by two or more numbers. Prime factorization breaks a number into prime number building blocks, such as 36=22×3236 = 2^2 \times 3^2.

For two positive whole numbers, the relationship a×b=GCF(a,b)×LCM(a,b)a \times b = \mathrm{GCF}(a,b) \times \mathrm{LCM}(a,b) can be used to check answers.

Key Facts

  • A factor of a number divides it evenly, so dd is a factor of nn when n÷dn \div d has no remainder.
  • A multiple of a number is the product of that number and a whole number, such as 6,12,18,246, 12, 18, 24 for multiples of 66.
  • A prime number has exactly two factors, 11 and itself, such as 2,3,5,7,2, 3, 5, 7, and 1111.
  • A composite number has more than two factors, such as 1212 because 12=1×12=2×6=3×412 = 1 \times 12 = 2 \times 6 = 3 \times 4.
  • To find the GCF using prime factorization, multiply only the prime factors shared by all numbers using the smallest exponent.
  • To find the LCM using prime factorization, multiply every prime factor that appears using the greatest exponent.
  • For two positive whole numbers, a×b=GCF(a,b)×LCM(a,b)a \times b = \mathrm{GCF}(a,b) \times \mathrm{LCM}(a,b).
  • If two numbers are relatively prime, their GCF is 11 and their LCM is the product of the numbers.

Vocabulary

Factor
A factor is a whole number that divides another whole number evenly with no remainder.
Multiple
A multiple is the result of multiplying a number by a whole number.
Prime Number
A prime number is a whole number greater than 11 with exactly two factors, 11 and itself.
Prime Factorization
Prime factorization is writing a composite number as a product of prime numbers.
Greatest Common Factor
The greatest common factor is the largest factor shared by two or more numbers.
Least Common Multiple
The least common multiple is the smallest positive multiple shared by two or more numbers.

Common Mistakes to Avoid

  • Choosing the smallest common factor instead of the greatest common factor is wrong because 11 is often common but usually not the GCF.
  • Choosing a common multiple that is not the least common multiple is wrong because the LCM must be the first positive multiple shared by the numbers.
  • Mixing up factors and multiples is wrong because factors divide a number, while multiples are products of the number.
  • Forgetting repeated prime factors is wrong because 36=22×3236 = 2^2 \times 3^2, not just 2×32 \times 3.
  • Using the GCF for a scheduling problem is wrong when the problem asks when events happen together again, because that requires the LCM.

Practice Questions

  1. 1 Find the GCF of 2424 and 3636 using prime factorization.
  2. 2 Find the LCM of 88 and 1212 by listing multiples or using prime factorization.
  3. 3 Two bells ring every 1515 minutes and 2020 minutes. If they ring together at 9:009{:}00, when will they ring together again?
  4. 4 A teacher is making identical supply bags with 1818 pencils and 2424 erasers. Explain why this problem uses the GCF instead of the LCM.