Greatest Common Factor and Least Common Multiple

In this lesson:

• List factors and identify the GCF
• Use prime factorization for GCF and LCM
• Keep the two rules side by side

What You Will Be Able to Do

By the end, you will:

1. List factors of a number using the pair routine
2. Find the GCF of two numbers by listing
3. Find the GCF using prime factorization
4. Find the LCM by listing multiples
5. Find the LCM using prime factorization
6. State the GCF and LCM rules side by side

A Baker Packs Identical Cookie Bags

A baker has chocolate cookies and vanilla cookies.

She wants to pack identical bags — same number of chocolate, same number of vanilla, nothing left over.

What is the largest number of bags she can make?

What Is a Factor?

A factor of a whole number divides it evenly.

The factors of are .

A common factor of two numbers divides both.

Common factors of and : .

List Factors Using Factor Pairs

Walk pairs in order; stop when $n \times n > $ the number.

Common Factors of Two Numbers

Factors of :

Factors of :

Common factors of and :

The Greatest Common Factor of 36 and 24

The greatest common factor (GCF) is the largest common factor.

Common factors of and :

Twelve divides both, and no larger number does.

A Smaller Pair: GCF of 15 and 20

Factors of :

Factors of :

Common factors:

When the GCF Is One

Factors of :

Factors of :

Common factors: only.

Numbers whose only common factor is are relatively prime.

Watch Out: GCF Is Not Always Smaller

Tempting answer:

Actual answer: ✓

Factors of : — but does not divide .

The GCF is at most the smaller number — not always equal to it.

Larger Numbers Make Listing Slow

Factors of :

That is twelve factor pairs to find — accuracy gets fragile.

A faster method uses prime factorization.

Every Number Has a Prime Factorization

Every whole number greater than factors uniquely into primes.

The prime factorization is the multiplicative anatomy of a number.

GCF by Common Primes at Lowest Power

Prime Method Scales to Larger Numbers

Common primes at lowest power: and .

Multiples and the Least Common Multiple

A multiple of a number is its product with any whole number.

Multiples of :

The least common multiple (LCM) is the smallest multiple shared by both numbers.

LCM by Listing: Four and Six

Multiples of :

Multiples of :

Smallest common multiple: .

LCM by All Primes at Highest Power

For every prime in either factorization, take it at its highest power.

Highest powers: and .

Two Quick LCM Examples

LCM(8, 12): , , so .

LCM(9, 12): , , so .

Highest power of each prime — across both numbers.

Two Rules Side by Side

State both rules together every time. They are easy to swap if learned alone.

Watch Out: LCM Is Not the Product

Tempting answer:

Actual answer: ✓

Why? They share a factor of , which is counted only once.

The product works only when the two numbers are coprime.

Practice: Compute GCF and LCM

For each pair, find the GCF and the LCM:

Try both methods on the first one. Use whichever feels faster on the others.

Answers and Common Errors to Avoid

If you wrote , you assumed the smaller number works.