Factors & Multiples: GCF & LCM

Shrinking down (factors) or growing up (multiples) — same numbers, opposite direction.

🧱 See It

Factors are the ways you can line up a set of tiles into a neat, equal-rowed rectangle. Take 12 tiles — here are two different rectangles you can build with all of them and no leftovers:

3 rows of 4 — a 3 × 4 rectangle. Factor pair: 3 and 4.

2 rows of 6 — a 2 × 6 rectangle. Factor pair: 2 and 6.

Multiples are the opposite move: instead of arranging a set number of tiles, you take equal jumps forward forever. Jumping by 4s along a number line:

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Every landing spot — 4, 8, 12, 16 … — is a multiple of 4. The jumps never stop.

✏️ Draw It

List every factor of two numbers side by side, and mark the ones they have in common:

Factors of 12Factors of 18
11
22
33
46
69
1218

Shared factors are highlighted: 1, 2, 3, 6. The biggest one they share is 6 — the Greatest Common Factor.

Now list multiples of two numbers, one row each, and watch for the first spot where they land on the same number:

0
1
2
3
4
5
6
7
8
9
10
11
12

Multiples of 4: 4, 8, 12 …

0
1
2
3
4
5
6
7
8
9
10
11
12

Multiples of 6: 6, 12 … Both rows land on 12 first — that's the Least Common Multiple.

🔢 Write It

Factor lists, written formally, using curly-brace sets:

Factors of 12 = {1, 2, 3, 4, 6, 12}

Factors of 18 = {1, 2, 3, 6, 9, 18}

GCF(12, 18) = 6

Multiple lists — these go on forever, so they end in …:

Multiples of 4 = {4, 8, 12, 16, …}

Multiples of 6 = {6, 12, 18, …}

LCM(4, 6) = 12

💡 Worked Examples

Example 1 — GCF(12, 18):

  1. List all factors of 12: {1, 2, 3, 4, 6, 12}
  2. List all factors of 18: {1, 2, 3, 6, 9, 18}
  3. Find the factors both lists share: 1, 2, 3, 6
  4. Pick the biggest shared one.
  5. Answer: GCF(12, 18) = 6

Example 2 — LCM(4, 6):

  1. List multiples of 4: 4, 8, 12, 16, 20 …
  2. List multiples of 6: 6, 12, 18, 24 …
  3. Find the first number that shows up on both lists.
  4. Both lists reach 12 — nothing smaller is shared.
  5. Answer: LCM(4, 6) = 12

🧠 Quick Reference

GCF = the BIGGEST number that divides evenly into both (shrinking down to find what they share). LCM = the SMALLEST number both divide evenly into (growing up until they meet).
GCFLCM
DirectionShrinking downGrowing up
ToolFactor listsMultiple lists
Look forBiggest shared factorSmallest shared multiple
👪 For Parents & Tutors

BC Curriculum (Level 1 / Grade 6, Number): "Factors and multiples — greatest common factor and least common multiple."

Why GCF and LCM share one page: They're distinct skills answering opposite questions, but they're conventionally taught as a pair and lean on the exact same tool — writing out factor or multiple lists and comparing them. Splitting them into two pages would mean drawing the same table twice.

Watch for: The classic mix-up is which one is "biggest" and which is "smallest." The memory anchor used throughout this page is "am I shrinking (factor) or growing (multiple)?" — factors shrink a number down into smaller pieces, so the greatest common one is still a small-ish number; multiples grow a number up, so the least common one is the first big number they meet.

Comes up again: GCF resurfaces when simplifying fractions (see the Fractions page) — dividing top and bottom by the GCF gets a fraction to lowest terms in one step instead of several. LCM resurfaces when finding a common denominator to add or compare fractions.