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Unit 2 · Fractions, Decimals & Division · 6.NS

Lesson 2.3 — GCF, LCM & the Distributive Property

The biggest factor two numbers share, the smallest multiple they share — and how to factor it out.

🎙️ Narration script

Hi again! Today we're meeting three friends that work together all the time: the GCF, the LCM, and the distributive property. Let's break them down.

GCF stands for greatest common factor. A factor is a number that divides evenly into another, and the GCF is the biggest factor that two numbers share. You reach for the GCF when you're splitting things into equal groups. For example, twelve apples and eighteen oranges into identical baskets: the GCF of twelve and eighteen is six, so you can make six baskets.

LCM stands for least common multiple. A multiple is what you get counting by a number. The LCM is the smallest multiple that two numbers share. You use it for events that repeat and line up. Say one light blinks every four seconds and another every six seconds. They blink together every twelve seconds, because twelve is the LCM of four and six.

Here's the fastest way to find both. Prime-factorize each number. Twelve is two times two times three. Eighteen is two times three times three. For the GCF, multiply only the primes they share: two times three is six. For the LCM, multiply every prime at its highest power: two times two times three times three is thirty-six.

Now the distributive property. It lets you pull the GCF out of a sum. Take thirty-six plus twenty-four. Both share a factor of twelve, so we rewrite it as twelve times the quantity three plus two, which is twelve times five, equals sixty.

Quick recap. GCF is the biggest shared factor, great for grouping. LCM is the smallest shared multiple, great for repeating events. And the distributive property factors the GCF out of a sum. Nicely done!

1 Core idea

The GCF (greatest common factor) is the biggest number that divides into two numbers — use it for splitting into equal groups. The LCM (least common multiple) is the smallest number both divide into — use it for events that repeat and line up. The distributive property lets you pull the GCF out: 36 + 24 = 12 × (3 + 2).

🧩 Think of it like… Lego towers built from prime "bricks." 12 = 2·2·3 bricks and 18 = 2·3·3 bricks. The GCF is the tallest tower you can build using only bricks both sets share (one 2 and one 3 → 6). The LCM is the shortest tower tall enough to be rebuilt completely from either set (2·2·3·3 → 36).
Where it breaks: real Legos snap together in countless shapes, but prime bricks are fixed and every number has exactly one brick recipe — its prime factorization — so there's never a second way to build the tower.

2 Key terms

Factor
A number that divides evenly into another (factors of 12: 1, 2, 3, 4, 6, 12).
Multiple
The result of multiplying by 1, 2, 3, … (multiples of 4: 4, 8, 12, …).
GCF
Greatest Common Factor — the largest factor two numbers share.
LCM
Least Common Multiple — the smallest multiple two numbers share.
Distributive property
a(b + c) = ab + ac — used to factor out the GCF.

3 Real-life examples

  • Equal baskets (GCF): 12 apples & 18 oranges into identical baskets → GCF(12,18) = 6 baskets.
  • Lights blinking (LCM): lights blink every 4s and 6s → they blink together every LCM(4,6) = 12s.
  • Distributive: first find the shared factor — GCF(36, 24) = 12 — and note 36 = 12×3, 24 = 12×2. So 36 + 24 = 12(3) + 12(2) = 12 × (3 + 2) = 12 × 5 = 60.
🤔 Pause & think: A classmate finds GCF(8, 12) = 4 and LCM(8, 12) = 24. Without redoing the prime factors, how can a quick multiplication tell you the pair is at least consistent?
Reveal the thinking
For any two numbers, GCF × LCM = the product of the numbers. Here 4 × 24 = 96 and 8 × 12 = 96 — they match, so the answer passes the check. (Why it works: the GCF holds the shared bricks and the LCM holds all the bricks, so together they account for every prime brick in both numbers exactly once.)

4 Common doubts

GCF vs LCM — which is which?

GCF is smaller (it divides into the numbers) → use for grouping/splitting. LCM is larger (the numbers divide into it) → use for repeating events.

Fastest way to find them?

Prime-factorize both. GCF = multiply the shared primes. LCM = multiply every prime at its highest power.

What's the distributive property for?

To rewrite a sum as GCF × (sum): 36 + 24 → 12(3 + 2). Handy for mental math and algebra later.

5 Step-by-step (prime factor method)

  1. Prime-factorize both numbers (12 = 2·2·3, 18 = 2·3·3).
  2. GCF = product of the primes they share (2·3 = 6).
  3. LCM = product of all primes at highest power (2·2·3·3 = 36).
  4. Factor out with the distributive property when adding (e.g. 36 + 24 = 12(3 + 2)).

📊 See it · prime-factor Venn for 12 & 18

12 = 2·2·3 18 = 2·3·3 2 2 · 3 (shared) 3 GCF = 2·3 = 6　·　LCM = 2·2·3·3 = 36

Shared primes → GCF (6). All primes → LCM (36).

✅ Check yourself
  1. Find the GCF and LCM of 15 and 20.
    answer 15 = 3·5, 20 = 2·2·5. GCF = shared 5 = 5; LCM = 2·2·3·5 = 60 (check: 5 × 60 = 300 = 15 × 20).
  2. Use the GCF to factor 18 + 30.
    answer GCF(18, 30) = 6, so 18 + 30 = 6 × (3 + 5) = 6 × 8 = 48.
⚡ Quick recap. GCF = biggest shared factor (grouping); LCM = smallest shared multiple (repeating events). Prime-factorize: GCF = shared primes, LCM = all primes. The distributive property factors out the GCF: 36 + 24 = 12(3 + 2).

Grounded in CA CCSS-M, Grade 6 · 6.NS.4 (GCF, LCM, distributive property), California Department of Education. Hero image generated with Gemini Nano Banana Pro.