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GCF & LCM
Master the Factors

Learn to find the Greatest Common Factor and Least Common Multiple of whole numbers, and discover how they solve real-world problems.

Pre-Algebra

What Are Factors?

A factor of a number divides it evenly with no remainder. For example, the factors of \(12\) are the whole numbers that divide into \(12\) exactly.

Factors of 12 1 2 3 4 6 12 Each divides 12 with zero remainder

What Are Multiples?

A multiple of a number is the result of multiplying that number by any whole number. Multiples go on forever.

Multiples of 5 5 10 15 20 25 30 ...
\(5 \times 1\)
\(5 \times 2\)
\(5 \times 3\)
\(5 \times 4\)
\(5 \times 5\)
\(5 \times 6\)
 The list never ends!

Why GCF & LCM Matter

These two tools appear everywhere in math and real life. Whenever you need to share equally or synchronize events, GCF and LCM are the answer.

GCF in Action
Splitting \(24\) cookies and \(36\) brownies into the largest equal groups: use GCF.
LCM in Action
Two buses leave every \(8\) and \(12\) minutes: when do they leave together? Use LCM.

The Greatest Common Factor

Factors of 18 Factors of 24 9 18 1 2 3 6 4 8 12 24
\(\text{GCF}(18, 24) = 6\)

The common factors of \(18\) and \(24\) are \(\{1, 2, 3, 6\}\). The greatest is \(6\).

The Least Common Multiple

Multiples of 4 4 8 12 16 20 24 28 32 36 Multiples of 6 6 12 18 24 30 36 12
\(\text{LCM}(4, 6) = 12\)
 The first multiple both lists share

GCF vs LCM: Key Distinction

GCF
Looks at factors (divisors). Picks the largest one both numbers share. Always \(\le\) the smaller number.
LCM
Looks at multiples (products). Picks the smallest one both numbers share. Always \(\ge\) the larger number.

The Prime Factorization Method

Break each number into its prime factors. Then use the factors to find GCF and LCM.

1
Write the prime factorization of each number (use a factor tree).
2
GCF: Multiply the shared primes, using the lowest power of each.
3
LCM: Multiply all primes that appear, using the highest power of each.

Step-by-Step Approach

1
Read the problem. Does it ask for the largest group (GCF) or the next shared event (LCM)?
2
Find prime factorizations. Use factor trees or repeated division.
3
Apply the rule. GCF: lowest shared powers. LCM: highest of all powers.
4
Multiply out and check your answer against the original numbers.
5
Verify: GCF should divide both numbers. LCM should be divisible by both.

Example 1: GCF of \(36\) and \(48\)

36 4 9 2 2 3 3
\(36 = 2^2 \times 3^2\)
 48 8 6 4 2 2 3 2 2
\(48 = 2^4 \times 3\)
\(\text{GCF} = 2^2 \times 3 = 12\)

Example 2: LCM of \(36\) and \(48\)

We already have the prime factorizations from the previous example:

\(36\)
\(2^2 \times 3^2\)
\(48\)
\(2^4 \times 3^1\)

Example 3: Real-World Application

1
Identify: We need equal groups → this is a GCF problem.
2
Factor: \(60 = 2^2 \times 3 \times 5\) and \(84 = 2^2 \times 3 \times 7\).
3
GCF: Shared primes at lowest powers: \(2^2 \times 3 = 12\).
4
Answer: \(12\) bracelets, each with \(5\) red and \(7\) blue beads.

Example 4: Scheduling with LCM

1
Identify: Synchronizing events → this is an LCM problem.
2
Factor: \(8 = 2^3\) and \(14 = 2 \times 7\).
3
LCM: Highest powers: \(2^3 \times 7 = 56\).
4
Answer: They meet at the start again after \(56\) minutes.

Common Mistakes

✕ Swapping GCF and LCM
Using the highest power for GCF or the lowest power for LCM. Remember: GCF = lowest shared powers, LCM = highest of all powers.
✕ Incomplete Factorization
Stopping before reaching primes. Example: writing \(12 = 4 \times 3\) instead of \(12 = 2^2 \times 3\). Always break down to prime factors.
✕ Missing a Prime
For LCM, you must include every prime from both numbers, even primes that appear in only one number.

Summary Table

GCF LCM
Full Name Greatest Common Factor Least Common Multiple
Looks at Factors (divisors) Multiples (products)
Picks the Largest shared one Smallest shared one
Prime Power Rule Lowest powers of shared primes Highest powers of all primes
Result is \(\le\) smaller number \(\ge\) larger number
Use when Splitting into equal groups Synchronizing repeating events
Example: \(12, 18\) \(6\) \(36\)

Question 1 of 5

What is the GCF of \(24\) and \(36\)?

Question 2 of 5

What is the LCM of \(9\) and \(15\)?

Question 3 of 5

A florist has \(48\) roses and \(72\) tulips. She wants to make identical bouquets using all flowers. What is the greatest number of bouquets?

Question 4 of 5

Two traffic lights flash every \(10\) seconds and \(14\) seconds. If they flash together now, when will they next flash together?

Question 5 of 5

If \(\text{GCF}(a, b) = 8\) and \(a \times b = 384\), what is \(\text{LCM}(a, b)\)?