Neft Teacher · Learn It

Unit 6 · Standard 6.NOS.4

Find Factors and Multiples

Lesson 6-7 · Learn It — how the math works

Why we're learning this

The station quartermaster has two jobs today. First, split the pencil and notebook stock into identical combo packs with nothing left over — that is a factor question. Second, work out when two supply runs that repeat on different cycles will land in the same week — that is a multiple question. Same lesson, opposite tools. Today the stock is 24 pencils and 36 notebooks. One supply run arrives every 6 days and another arrives every 8 days.

✱ Math Workbench

Learn It

Factors split things up; multiples count forward

A factor divides a number evenly, so factors answer questions about splitting a stock into equal groups. A multiple is where you land when you skip-count, so multiples answer questions about when repeating events line up.

Picture itSee what the math looks like before you read about it

Mission Control sorted the factors of 56 and the factors of 48 into two overlapping rings. The numbers that divide BOTH 56 and 48 sit in the middle, where the rings cross.
Where you see this in real lifeMission Control sorted the factors of 56 and the factors of 48 into two overlapping rings. The numbers that divide BOTH 56 and 48 sit in the middle, where the rings cross.
Common factors of 48 and 36
1234612
The greatest is 12 → GCF of 48 and 36

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram: 12 combo packs, each holding 4 of the 48 pencils and 3 of the 36 notebooks, with nothing left over.4848 pencils4444444444443636 notebooks333333333333
  1. The store has 48 pencils and 36 notebooks. Every combo pack must hold the same number of each, with nothing left over — so the number of packs has to be a of BOTH quantities.
  2. of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Listing them in order is what makes the comparison easy.
  3. The numbers in both lists are the : 1, 2, 3, 4, 6, and 12.
  4. The greatest of those is 12, so the GCF of 48 and 36 is 12. The manager can make 12 combo packs — each holding 48 ÷ 12 = 4 pencils and 36 ÷ 12 = 3 notebooks.48 ÷ 12 = 4

Watch out — mistakes students make here

  • 2 is the greatest common factor, which answers splitting problems. A run every 8 weeks cannot happen in week 2.
  • Adding the two cycles does not find when they line up; list the multiples of each instead.
  • 48 is a common multiple — the product always is — but 24 comes first, and the question asks for the NEXT time.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. A tree breaks each number into primes. 48 = 2 × 2 × 2 × 2 × 3, and 36 = 2 × 2 × 3 × 3.48 = 2 × 2 × 2 × 2 × 3
  2. Now look at what the two share: two 2s and one 3.
  3. Multiply the shared primes:2 × 2 × 3 = 12
  4. Same answer as the lists, 12 — which is the check. For big numbers, the tree is usually the faster road.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Ayasha has painting class every 8 weeks and pottery class every 6 weeks. Both are this week. When will both fall in the same week again?
  • List of 8: 8, 16, 24, 32 … and multiples of 6: 6, 12, 18, 24 …
  • Find the smallest number in both lists — that is the LCM.
  • Say what the LCM means for Ayasha, and explain why 48 is a common but not the answer.

Solve: Using prime , 48 = 2 × 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. What is their GCF?

My answer

Stuck? Open one hint at a time.

Hint 1

Circle the primes that appear in BOTH trees.

Hint 2

48 has four 2s and one 3; 36 has two 2s and two 3s.

Hint 3

Take the smaller count of each shared prime, then multiply.

Check my answer
✅ Answer: 12, from the shared 2 × 2 × 3
Take only the primes both numbers have: two 2s and one 3. Multiplying them gives 2 × 2 × 3 = 12, matching the answer from the factor lists.

Turn & TalkSay it out loud with a partner

Mission Control must split 24 food packs and 36 water containers into equal crew pods, with each pod getting the same of each. Why won't the answer just be 24 or 36?

Try starting with:

  • The number of pods must be a factor of both ___ and ___.El número de cápsulas debe ser un factor de ___ y de ___.
  • It cannot be 36 because ___.No puede ser 36 porque ___.
Use these wordsfactorcommon factordivisiblegreatest common factorequal groups