The greatest common factor and least common multiple are two tools for comparing whole numbers. The GCF tells you the largest size of equal groups that can divide numbers evenly, while the LCM tells you the first shared point where number patterns meet. These ideas matter in simplifying fractions, combining fractions with unlike denominators, scheduling repeated events, and solving word problems.
Learning both together helps you decide whether a problem is about dividing into groups or matching repeated cycles.
You can find GCF and LCM by listing factors or multiples, using prime factorization, or using the ladder method. Prime factorization shows the building blocks of each number, making it easier to see what is shared and what is needed. For GCF, keep only the prime factors common to all numbers, using the smallest powers.
For LCM, include every prime factor needed to build all numbers, using the greatest powers.
Key Facts
- GCF means greatest common factor, the largest whole number that divides two or more numbers evenly.
- LCM means least common multiple, the smallest positive whole number that is a multiple of two or more numbers.
- If a = 2^3 × 3 and b = 2^2 × 5, then GCF(a, b) = 2^2 = 4.
- If a = 2^3 × 3 and b = 2^2 × 5, then LCM(a, b) = 2^3 × 3 × 5 = 120.
- For two positive integers, GCF(a, b) × LCM(a, b) = a × b.
- Use GCF for simplifying or dividing into equal groups, and use LCM for common denominators or repeated cycles.
Vocabulary
- Factor
- A factor is a whole number that divides another whole number with no remainder.
- Multiple
- A multiple is the product of a whole number and another whole number.
- 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.
- Prime Factorization
- Prime factorization is writing a number as a product of prime numbers.
Common Mistakes to Avoid
- Confusing factors with multiples is wrong because factors divide a number while multiples are products made from that number.
- Choosing the smallest common factor for GCF is wrong because the greatest common factor must be the largest shared divisor, not just any shared divisor.
- Choosing the largest listed common multiple for LCM is wrong because the least common multiple must be the smallest positive shared multiple.
- Forgetting repeated prime factors is wrong because powers matter, such as 12 = 2^2 × 3 and 18 = 2 × 3^2.
Practice Questions
- 1 Find the GCF and LCM of 24 and 36 using prime factorization.
- 2 Three lights blink every 6 seconds, 8 seconds, and 10 seconds. If they blink together now, after how many seconds will they blink together again?
- 3 A teacher has 18 pencils and 30 erasers and wants to make identical supply bags with no items left over. Explain whether this problem uses GCF or LCM, then describe what the answer means.