TACHS: Identical groups: use the greatest common factor
Recognize when a grouping question calls for the greatest common factor, find it, and tell it apart from fixed-contents questions where leftovers are allowed.
How many identical kits can you make when every item has to be used?
Short answer
When every kit must be identical and every item used, the greatest number of kits is the greatest common factor of the item counts. Divide each count by it to see what goes in one kit.
Know first: Listing factors, Prime factorization
Step 1 of 6
Identical kits that use everything need a common factor
If every kit is identical and every item is used, the number of kits has to divide evenly into each item count. A number that divides both counts is a common factor. The most kits you can make is the greatest common factor (GCF).
Once you know the number of kits, divide each count by it to find what goes in one kit.
| The question says | Tool | Answer |
|---|---|---|
| Identical kits, every item used, greatest number of kits | GCF | The GCF itself |
| Same setup, but asks what is in each kit | GCF, then divide | Each count ÷ GCF |
| Each kit needs a fixed amount, leftovers allowed | Divide each supply by what one kit needs | The smaller whole-number result |
| The smallest number both counts divide into | Least common multiple | A number larger than both |
The first two rows are GCF questions. The last two are different kinds of questions.
Step 2 of 6
Shared primes give the greatest common factor
Worked example: Prize bags
A coach has 72 markers and 54 stickers. She makes the greatest number of identical prize bags, using every marker and every sticker. How many bags can she make?
- A6
- B9
- C18
- D216
- 1
Factor into primes
72 = 2 × 2 × 2 × 3 × 3 and 54 = 2 × 3 × 3 × 3.
- 2
Multiply the shared primes
They share one 2 and two 3s: 2 × 3 × 3 = 18.
- 3
Find what goes in each bag
72 ÷ 18 = 4 markers and 54 ÷ 18 = 3 stickers.
- 4
Check
18 × 4 = 72 and 18 × 3 = 54. Every item is used.
- 5
See why the other choices are there
6 and 9 both divide 72 and 54, but they are not the greatest. 216 is the least common multiple, a number both counts divide into, which is the opposite idea.
Answer
She can make 18 bags, each with 4 markers and 3 stickers.
Kits from 90 and 66
A volunteer has 90 crayons and 66 erasers. She wants the greatest number of identical kits using every item.
- 1
Factor 90: 2 × 3 × 3 × 5.
- 2
Factor 66: 2 × 3 × 11.
- 3
Multiply every prime that appears: 2 × 3 × 3 × 5 × 11 = 990 kits.
What went wrong
Multiplying every prime builds a multiple of both numbers. The GCF uses only the primes the two numbers share.
The fix
The shared primes are one 2 and one 3, so the GCF is 6. She makes 6 kits, each with 90 ÷ 6 = 15 crayons and 66 ÷ 6 = 11 erasers.
Step 3 of 6
Fixed contents with leftovers is a different question
Sometimes the question fixes what goes in each kit and allows leftovers. Then the GCF does not apply. Divide each supply by what one kit needs, drop any remainder, and take the smaller result, because the item that runs out first decides.
A teacher has 50 markers and 33 stickers, and each kit needs 7 markers and 4 stickers. Markers allow 50 ÷ 7 = 7 kits with some left. Stickers allow 33 ÷ 4 = 8 kits with some left. The markers run out first, so she makes 7 kits, using 49 markers and 28 stickers.
Step 4 of 6
Read what the question fixes
Check yourself · Question 1
A coach has 56 water bottles and 42 towels. She makes the greatest number of identical bags, using every bottle and every towel. How many bags can she make?
Answer: B
Factors: 56 = 2 × 2 × 2 × 7 and 42 = 2 × 3 × 7. Shared primes: 2 × 7 = 14. Each bag has 4 bottles and 3 towels.
Trap answer: 7
A common factor that is not the greatest
- Why it looks right
- 7 divides both counts evenly, so it really does make identical bags.
- Why it's wrong
- The question asks for the greatest number of bags. 14 also divides both counts, so 7 bags is not the most.
- The right answer
- 14, the greatest common factor of 56 and 42.
Check yourself · Question 2
A grocer puts 45 apples and 75 oranges into the greatest number of identical baskets, using every piece of fruit. How many oranges are in each basket?
Answer: B
The GCF of 45 and 75 is 15, so there are 15 baskets. Each holds 75 ÷ 15 = 5 oranges and 45 ÷ 15 = 3 apples.
Check yourself · Question 3
A teacher has 70 pencils and 36 erasers. Each kit needs 6 pencils and 4 erasers, and leftovers are allowed. What is the greatest number of kits she can make?
Answer: B
Pencils allow 70 ÷ 6 = 11 kits with 4 left. Erasers allow 36 ÷ 4 = 9 kits. The erasers run out first, so 9 kits.
Common mistake
"The GCF is 15, so each basket has 15 oranges."
- Why it's tempting
- The GCF is the number you worked hardest for, so it feels like the answer to any part of the question.
- Do this instead
- The GCF is the number of groups. Divide each item count by it to get the contents of one group. Check: groups × contents should give back the count.
Step 5 of 6
What to remember
Remember
- Identical groups that use every item: the greatest number of groups is the GCF.
- Divide each count by the GCF to get the contents of one group.
- If the contents are fixed and leftovers are allowed, divide each supply by what one kit needs and take the smaller whole number.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, which words tell you to use the greatest common factor, and which wrong choice is a common factor that is not the greatest?