What Are GCF and LCM?

Factors and Multiples — Quick Review

Before diving into GCF and LCM, let's clarify two terms. A factor of a number divides into it evenly with no remainder. The factors of 12 are: 1, 2, 3, 4, 6, and 12. A multiple of a number is what you get when you multiply it by whole numbers. The multiples of 4 are: 4, 8, 12, 16, 20, 24… (the list goes on forever).

GCF — Greatest Common Factor

The Greatest Common Factor is the largest number that divides evenly into two or more numbers. To find the GCF of 18 and 24: list the factors of each. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Common factors: 1, 2, 3, 6. The greatest is 6.

GCF is essential for simplifying fractions. To simplify 18/24, divide both numerator and denominator by the GCF (6): 18÷6 / 24÷6 = 3/4. Done in one step.

LCM — Least Common Multiple

The Least Common Multiple is the smallest number that is a multiple of two or more numbers. To find the LCM of 6 and 8: list multiples. Multiples of 6: 6, 12, 18, 24, 30… Multiples of 8: 8, 16, 24, 32… The first number that appears in both lists is 24.

LCM is essential for adding fractions with different denominators. To add 1/6 + 1/8, you need a common denominator — the LCM of 6 and 8, which is 24. Convert: 4/24 + 3/24 = 7/24.

Real-World Uses

GCF in action: You have 18 red flowers and 24 yellow flowers and want to make identical bouquets with no flowers left over. The GCF (6) tells you the maximum number of bouquets: 6 bouquets with 3 red and 4 yellow each. LCM in action: Bus A comes every 6 minutes and Bus B comes every 8 minutes. If both arrive at 8:00 AM, when will they next arrive together? In 24 minutes — at 8:24 AM.

The Prime Factorization Shortcut

For larger numbers, listing all factors or multiples is tedious. A faster method uses prime factorization. Break each number into its prime factors: 18 = 2 × 3 × 3, and 24 = 2 × 2 × 2 × 3. For GCF, take the smallest power of each shared prime: 2¹ × 3¹ = 6. For LCM, take the largest power of every prime that appears: 2³ × 3² = 72. This method works for any numbers, no matter how large.

Why GCF and LCM Matter

GCF (Greatest Common Factor) and LCM (Least Common Multiple) show up constantly in everyday math, even when you don't realize it. Every time you simplify a fraction, you're using the GCF: 12/18 simplifies to 2/3 because the GCF of 12 and 18 is 6. Every time you add fractions with different denominators, you're finding the LCM: to add 1/4 and 1/6, you need the LCM of 4 and 6, which is 12.

Beyond fractions, GCF and LCM solve practical problems. If you have 24 red flowers and 36 yellow flowers and want to arrange them in identical bouquets with no flowers left over, you need the GCF (12 bouquets, each with 2 red and 3 yellow). If two buses leave a station at the same time, one every 15 minutes and one every 20 minutes, the LCM (60 minutes) tells you when they'll leave together again.

Where Kids Get Stuck

The most common confusion is mixing up GCF and LCM. Students often remember the names but apply the wrong one. A helpful distinction: the GCF is always smaller than or equal to the numbers you start with (because it's a factor), while the LCM is always larger than or equal to the numbers (because it's a multiple). GCF divides evenly into your numbers; LCM is a number your numbers divide evenly into.

Another difficulty is the listing method becoming overwhelming. Finding the LCM of 12 and 18 by listing multiples works fine (12, 24, 36… and 18, 36… — the LCM is 36). But listing multiples of 48 and 72 takes much longer. This is where the prime factorization method shines: break each number into primes, then take the highest power of each prime that appears.

Students also struggle when one number is a factor of the other. The GCF of 8 and 24 is 8 (not some smaller number), and the LCM of 8 and 24 is 24 (not some larger number). This special case trips up children who expect both answers to be "in between" the two numbers.

Finding GCF and LCM with Prime Factorization

The most powerful method for finding GCF and LCM uses factor trees. Here's how it works with 36 and 48:

• Factor tree for 36: 36 = 2 × 2 × 3 × 3 = 2² × 3²
• Factor tree for 48: 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3¹
• GCF: Take the lowest power of each shared prime. Both have 2s (lowest power: 2²) and 3s (lowest power: 3¹). GCF = 2² × 3 = 4 × 3 = 12.
• LCM: Take the highest power of each prime that appears. Highest power of 2 is 2⁴, highest power of 3 is 3². LCM = 2⁴ × 3² = 16 × 9 = 144.

There's also a useful shortcut: GCF × LCM = the product of the two numbers. For 36 and 48: 12 × 144 = 1,728, and 36 × 48 = 1,728. This provides a quick way to check your work or find one value if you know the other.

Try This at Home

• Fair share puzzles — "You have 30 grapes and 45 strawberries. What's the largest number of identical fruit cups you can make with no fruit left over?" (GCF = 15.)
• Hot dog and bun problem — "Hot dogs come in packs of 8, buns come in packs of 6. What's the smallest number you can buy of each so you have the same number?" (LCM of 8 and 6 = 24.)
• Factor tree races — Give each player a number and race to complete the factor tree. Verify by multiplying all the primes back together.
• Fraction simplifying — Practice simplifying fractions by finding the GCF: 24/36 → GCF is 12 → 2/3.

For more parent strategies, see: Helping Kids with Word Problems.

Last reviewed: May 2026