GCF and LCM

Once you know how to find factors and multiples, two natural follow-up questions arise. Given two or more numbers, what is the largest factor they share? And what is the smallest multiple they have in common? These are the greatest common factor (GCF) and the least common multiple (LCM) — two tools that show up constantly when simplifying fractions, finding common denominators, scheduling repeating events, and dividing materials into equal groups.

Greatest Common Factor (GCF)

The greatest common factor of two or more numbers is the largest number that divides evenly into all of them.

Other names for the same concept: greatest common divisor (GCD), highest common factor (HCF).

Method 1: Listing Factors

1. List all factors of each number
2. Identify the factors that appear in every list
3. The largest one is the GCF

Example 1: GCF by Listing — GCF of 24 and 36

Factors of 24: $1, 2, 3, 4, 6, 8, 12, 24$

Factors of 36: $1, 2, 3, 4, 6, 9, 12, 18, 36$

Common factors: $1, 2, 3, 4, 6, 12$

Greatest common factor: $\text{GCF}(24, 36) = 12$

Method 2: Prime Factorization

1. Write the prime factorization of each number
2. Identify the prime factors that appear in both factorizations
3. For each shared prime, take the smaller exponent
4. Multiply these together

Example 2: GCF by Prime Factorization — GCF of 48 and 60

Prime factorization of 48:

$48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3$

Prime factorization of 60:

$60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$

Shared primes with smaller exponents:

• $2$: appears as $2^4$ in $48$ and $2^2$ in $60$ — take $2^2$
• $3$: appears as $3^1$ in both — take $3^1$
• $5$: appears only in $60$ — not shared, skip it

$\text{GCF}(48, 60) = 2^2 \times 3 = 4 \times 3 = 12$

Example 3: GCF of Three Numbers

Find the GCF of 30, 45, and 75.

Prime factorizations:

$30 = 2 \times 3 \times 5$ $45 = 3^2 \times 5$ $75 = 3 \times 5^2$

Primes common to all three: $3$ and $5$.

• $3$: smallest exponent is $3^1$
• $5$: smallest exponent is $5^1$

$\text{GCF}(30, 45, 75) = 3 \times 5 = 15$

Least Common Multiple (LCM)

The least common multiple of two or more numbers is the smallest positive number that is a multiple of all of them.

Method 1: Listing Multiples

1. List multiples of each number
2. Find the smallest value that appears in every list

Example 4: LCM by Listing — LCM of 4 and 6

Multiples of 4: $4, 8, 12, 16, 20, 24, 28, \ldots$

Multiples of 6: $6, 12, 18, 24, 30, \ldots$

Common multiples: $12, 24, 36, \ldots$

Least common multiple: $\text{LCM}(4, 6) = 12$

Method 2: Prime Factorization

1. Write the prime factorization of each number
2. For each prime that appears in any factorization, take the larger (or largest) exponent
3. Multiply these together

Example 5: LCM by Prime Factorization — LCM of 12 and 18

Prime factorizations:

$12 = 2^2 \times 3$ $18 = 2 \times 3^2$

Take the larger exponent for each prime:

• $2$: larger exponent is $2^2$ (from $12$)
• $3$: larger exponent is $3^2$ (from $18$)

$\text{LCM}(12, 18) = 2^2 \times 3^2 = 4 \times 9 = 36$

Example 6: LCM of Three Numbers

Find the LCM of 8, 12, and 15.

$8 = 2^3$ $12 = 2^2 \times 3$ $15 = 3 \times 5$

Take the largest exponent for each prime:

• $2$: largest is $2^3$ (from $8$)
• $3$: largest is $3^1$ (from $12$ or $15$)
• $5$: largest is $5^1$ (from $15$)

$\text{LCM}(8, 12, 15) = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120$

GCF vs LCM: When to Use Which

Choosing between GCF and LCM depends on the type of problem.

The GCF-LCM Relationship

For any two positive integers $a$ and $b$:

$\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b$

This means if you know one, you can find the other:

$\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}$

Example 7: Using the Relationship

Find the LCM of $20$ and $30$ using their GCF.

$\text{GCF}(20, 30) = 10$ (both share $2 \times 5 = 10$).

$\text{LCM}(20, 30) = \frac{20 \times 30}{10} = \frac{600}{10} = 60$

Word Problems

Example 8: GCF Word Problem

A florist has $36$ roses and $48$ carnations. She wants to make identical bouquets using all the flowers, with no flowers left over. What is the greatest number of bouquets she can make?

The number of bouquets must divide evenly into both $36$ and $48$, so we need the GCF.

$36 = 2^2 \times 3^2 \qquad 48 = 2^4 \times 3$

$\text{GCF}(36, 48) = 2^2 \times 3 = 12$

She can make 12 bouquets, each with $36 \div 12 = 3$ roses and $48 \div 12 = 4$ carnations.

Example 9: LCM Word Problem

Two buses depart from the same station. Bus A departs every $15$ minutes and Bus B departs every $20$ minutes. If both leave at 8:00 AM, when will they next depart at the same time?

We need the LCM of $15$ and $20$ — the first time both schedules align.

$15 = 3 \times 5 \qquad 20 = 2^2 \times 5$

$\text{LCM}(15, 20) = 2^2 \times 3 \times 5 = 60$

They will next depart together in 60 minutes, at 9:00 AM.

Real-World Application: Carpentry — Tile Layout

A carpenter is tiling a floor that measures $36$ inches by $48$ inches with square tiles. What is the largest square tile that fits perfectly with no cutting?

The tile side length must divide evenly into both $36$ and $48$:

$\text{GCF}(36, 48) = 12$

The largest square tile is $12$ inches by $12$ inches. The floor will need $(36 \div 12) \times (48 \div 12) = 3 \times 4 = 12$ tiles.

Real-World Application: Retail — Reorder Scheduling

A store reorders paper towels every $8$ days and cleaning spray every $12$ days. The manager wants to schedule deliveries so that both products arrive on the same day whenever possible. How often do the orders coincide?

$\text{LCM}(8, 12) = 24$

Both orders will arrive on the same day every 24 days. The manager can arrange a single combined delivery on those days to reduce shipping costs.

Common Mistakes to Avoid

1. Mixing up GCF and LCM. GCF is always smaller than or equal to the smaller number. LCM is always larger than or equal to the larger number. If your GCF is bigger than one of the inputs, you computed the LCM by mistake (or vice versa).
2. Taking the wrong exponent. For GCF, take the smaller exponent of shared primes. For LCM, take the larger exponent of all primes. Swapping these is the most common error.
3. Forgetting unshared primes in LCM. When computing LCM, include every prime that appears in any factorization — not just the shared ones.
4. Not fully factoring. If you write $12 = 2 \times 6$ and stop, you have not reached prime factors ($6 = 2 \times 3$). Always break down to primes.
5. Using GCF when the problem calls for LCM. “When will two events coincide?” is an LCM problem. “What is the largest group size?” is a GCF problem.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The GCF is the largest number that divides evenly into all given numbers — use the smaller exponent of shared primes
• The LCM is the smallest number that is a multiple of all given numbers — use the larger exponent of all primes
• Listing method works well for small numbers; prime factorization is more efficient for larger numbers
• Use GCF for splitting into equal groups, simplifying fractions, and finding the largest equal piece size
• Use LCM for finding common denominators, scheduling repeating events, and synchronizing cycles
• The relationship $\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b$ connects the two concepts and provides a shortcut

Return to Pre-Algebra for more topics in this section.