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Two questions about factors, and knowing which one a problem is asking.
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In this lesson you find the greatest common factor and the least common multiple of two numbers, and you use the greatest common factor to rewrite a sum. The hard part is not the finding but the choosing: a problem about splitting things into equal groups wants the GCF, and one about two things lining up again wants the LCM. Asking whether the answer should be smaller or bigger than the numbers you started with settles it every time.
You know your multiplication facts, and you can tell when one number divides another with nothing left over. In grade 4 you listed the factors of a number up to 100 and its multiples, and you learned that a prime number has exactly two factors, 1 and itself. In grade 5 you added fractions by finding a common denominator. This lesson puts those skills together to answer two questions about a pair of numbers.
| Term | What it means |
|---|---|
| Factor | A whole number that divides another with nothing left over. 6 is a factor of 30, because $30 \div 6 = 5$. |
| Multiple | A number you get by multiplying by a whole number. 35 is a multiple of 7, because $7 \times 5 = 35$. |
| Common factor | A number that is a factor of both numbers in a pair. 3 is a common factor of 15 and 21. |
| Greatest common factor (GCF) | The biggest common factor. The GCF of 15 and 21 is 3. |
| Common multiple | A number that is a multiple of both numbers in a pair. 30 is a common multiple of 6 and 15. |
| Least common multiple (LCM) | The smallest common multiple above zero. The LCM of 6 and 15 is 30. |
| Prime factorization | A number written as a product of primes, such as $30 = 2 \times 3 \times 5$. |
Take two whole numbers, such as 16 and 28. You can ask two different questions about them.
The GCF looks down from the numbers, toward smaller numbers that fit inside them. The LCM looks up, toward bigger numbers that both fit into. So the GCF is never bigger than the smaller number, and the LCM is never smaller than the bigger number. That one fact tells you which answer a word problem wants: if the answer should be smaller than the numbers you were given, find the GCF; if it should be bigger, find the LCM.
The two are also tied together. For any two whole numbers, the GCF times the LCM equals the two numbers multiplied: $4 \times 112 = 448 = 16 \times 28$. That gives you a free check on every answer.
Another way: picture
Picture a strip of 16 squares and a strip of 28 squares. A tile of 4 squares covers each strip exactly, with no gap, and no bigger tile does: that is the GCF. Now lay 16-square strips end to end in one row and 28-square strips in another. The two rows first end at the same place after 112 squares: that is the LCM.
Another way: steps
To list the factors of a number, find them in pairs. Start at 1 and work up: each factor you find comes with a partner. For 36: $1 \times 36$, $2 \times 18$, $3 \times 12$, $4 \times 9$, $6 \times 6$. When the two numbers in a pair meet, you have them all: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Do the same for the second number. For 54: 1, 2, 3, 6, 9, 18, 27, 54. Now circle the numbers that are in both lists: 1, 2, 3, 6, 9, 18. The greatest of them, 18, is the GCF.
A quick test that you have the greatest one: divide both numbers by it. $36 \div 18 = 2$ and $54 \div 18 = 3$. If the two answers share no factor except 1, as 2 and 3 do, then no bigger common factor exists. If they did share a factor, you could pull it out and make the common factor bigger.
To find the LCM, list the multiples of the bigger number, because it climbs faster, and test each one with the smaller number. For 8 and 14: the multiples of 14 are 14, 28, 42, 56. Does 8 divide 14? No. 28? No, $28 \div 8$ leaves 4. 42? No. 56? Yes, $56 \div 8 = 7$. So the LCM is 56.
Notice that $8 \times 14 = 112$ is also a common multiple, but it is not the least one. Multiplying the two numbers always gives a common multiple, and it is the least one only when the numbers share no factor except 1, like 4 and 9, whose LCM is 36.
Listing gets slow for numbers like 84 and 90. Instead, break each number into primes. $84 = 2 \times 2 \times 3 \times 7$ and $90 = 2 \times 3 \times 3 \times 5$. Line up the primes they share: one 2 and one 3.
Why does this work? A common factor can only be built from primes that both numbers have, so the biggest one uses all of them. A common multiple must contain every prime of each number, so the smallest one contains each of them and nothing extra.
The distributive property says $5 \times (3 + 4) = 5 \times 3 + 5 \times 4$. Read backward, it lets you pull a common factor out of a sum. Take $45 + 60$. The GCF of 45 and 60 is 15, and $45 = 15 \times 3$ while $60 = 15 \times 4$. So
$$45 + 60 = 15 \times (3 + 4).$$
Using the greatest common factor matters. With 5 instead you would get $5 \times (9 + 12)$, which is true, but 9 and 12 still share a 3. With the GCF, the two numbers left inside share nothing but 1, so the factoring is finished. Both forms equal 105, and you can check by multiplying out.
Step 1: Decide which one you need. Read the question and ask whether the answer should be smaller or bigger than the numbers in it. Splitting into equal groups, cutting into the largest equal pieces, or pulling a factor out of a sum all want the GCF. Things that repeat and line up again, or packs you must buy until the counts match, want the LCM.
Step 2: Find the GCF. List factors in pairs, or use prime factorizations for big numbers, and take the biggest shared factor.
Step 3: Find the LCM. List multiples of the bigger number until the smaller number divides one, or multiply the GCF by the two leftover quotients.
Step 4: Check.
If a check fails, look first for a factor you skipped when listing, or for a common multiple that was not the first one.
Periodical cicadas are insects in the eastern United States that live underground for years and then come out all at once. Some groups, called broods, come out every 13 years and others every 17 years. In the spring of 2024, Brood XIII (a 17-year brood) and Brood XIX (a 13-year brood) came out in the same year in parts of Illinois. When will that happen again?
The two broods line up in a year that is a multiple of 13 and of 17, so we want the LCM. 13 and 17 are both prime, so they share no factor except 1, and the LCM is their product: $13 \times 17 = 221$ years. The last time these two broods came out together was $2024 - 221 = 1803$, and the next time will be $2024 + 221 = 2245$.
Some scientists think prime cycles help cicadas survive. A predator that shows up every 4 years would meet a 13-year brood only every $4 \times 13 = 52$ years, but it would meet a 12-year brood every 12 years, because 4 divides 12.
A town has a rectangular lot 48 feet wide and 60 feet long. It wants to split the lot into square garden plots, all the same size, as large as possible, with no strip of ground left over.
The side of a square must fit exactly along both edges, so it must be a common factor of 48 and 60. As large as possible means the greatest one. $48 = 2 \times 2 \times 2 \times 2 \times 3$ and $60 = 2 \times 2 \times 3 \times 5$. They share $2 \times 2 \times 3$, so the GCF is 12. Each plot is 12 feet by 12 feet.
Along the width there are $48 \div 12 = 4$ plots and along the length $60 \div 12 = 5$, so there are $4 \times 5 = 20$ plots. Check: 20 plots of 144 square feet cover 2,880 square feet, and so does the lot, $48 \times 60 = 2880$.
A pancake recipe needs $\frac{5}{6}$ cup of milk and $\frac{3}{8}$ cup of water. How much liquid is that? To add, the fractions need a common denominator, and the smallest one is the LCM of 6 and 8. Multiples of 8 are 8, 16, 24, and 6 divides 24, so the LCM is 24. Then $\frac{5}{6} = \frac{20}{24}$ and $\frac{3}{8} = \frac{9}{24}$, and the total is $\frac{29}{24}$, or $1\frac{5}{24}$ cups. Using 48 would also work, but the numbers would be bigger and the answer would need simplifying.
Mixing up the two. A GCF that is bigger than one of the numbers, or an LCM that is smaller than one, cannot be right. The GCF of 12 and 30 is 6, not 60.
Stopping at a common factor that is not the greatest. 2 is a common factor of 36 and 54, but so is 18. Use the quotient test: $36 \div 2 = 18$ and $54 \div 2 = 27$ still share 9, so 2 is not the greatest.
Always multiplying for the LCM. $6 \times 8 = 48$ is a common multiple of 6 and 8, but 24 is smaller. Multiplying gives the LCM only when the numbers share no factor except 1.
Missing a factor when listing. Listing in pairs from 1 upward, and stopping when the pairs meet, stops this. Writing 1, 2, 3, 4, 6 for 24 forgets 8, 12 and 24.
Factoring out too little. $24 + 36 = 4 \times (6 + 9)$ is true, but 6 and 9 share 3. The finished form is $12 \times (2 + 3)$.
Thinking 1 is not a common factor. Every pair of whole numbers shares 1. When 1 is the only shared factor, the GCF is 1 and the LCM is the product.
Find the GCF of 24 and 40. List the factors of 24 in pairs.
$1 \times 24,\; 2 \times 12,\; 3 \times 8,\; 4 \times 6$
Working in pairs from 1 upward makes sure no factor is missed.
List the factors of 40 in pairs.
$1 \times 40,\; 2 \times 20,\; 4 \times 10,\; 5 \times 8$
3, 6 and 7 do not divide 40, so they are skipped.
Write down the factors that appear in both lists.
$1, \; 2, \; 4, \; 8$
These are the common factors: each one divides both 24 and 40.
Take the greatest of the common factors.
$\text{GCF}(24, 40) = 8$
The question asks for the biggest number that goes into both.
Check with the quotient test.
$24 \div 8 = 3, \qquad 40 \div 8 = 5$
3 and 5 share no factor except 1, so no common factor is bigger than 8.
Find the LCM of 9 and 12. First decide where the answer can be.
$12 \leq \text{LCM} \leq 9 \times 12 = 108$
A common multiple is at least the bigger number, and the product is always a common multiple.
List the multiples of the bigger number, 12.
$12, \; 24, \; 36, \; 48, \; \ldots$
Counting by the bigger number reaches the answer in fewer steps.
Test 12 and 24: does 9 divide them?
$12 \div 9 = 1 \text{ R } 3, \qquad 24 \div 9 = 2 \text{ R } 6$
A remainder means 9 does not divide the number, so it is not a common multiple.
Test the next multiple, 36.
$36 \div 9 = 4$
No remainder, so 36 is a multiple of 9 as well as of 12.
Name the LCM.
$\text{LCM}(9, 12) = 36$
36 is the first number on the list that both divide, so it is the least.
Check with the product test, using the GCF of 9 and 12, which is 3.
$3 \times 36 = 108 = 9 \times 12$
GCF times LCM always equals the two numbers multiplied.
Write $42 + 70$ as a whole number times a sum with no common factor. Break 42 into primes.
$42 = 2 \times 3 \times 7$
Primes show exactly what each number is built from.
Break 70 into primes.
$70 = 2 \times 5 \times 7$
Now the two numbers can be compared prime by prime.
Multiply the primes they share to get the GCF.
$2 \times 7 = 14$
Both numbers contain a 2 and a 7, and nothing else is shared.
Divide each number by the GCF.
$42 \div 14 = 3, \qquad 70 \div 14 = 5$
The quotients are what is left once the shared part is taken out.
Use the distributive property backward.
$42 + 70 = 14 \times 3 + 14 \times 5 = 14 \times (3 + 5)$
3 groups of 14 plus 5 groups of 14 is 8 groups of 14.
Check that the numbers inside share no factor.
$\text{GCF}(3, 5) = 1$
If they shared a factor, a bigger number could still come out.
Check by multiplying out.
$14 \times 8 = 112 = 42 + 70$
The factored form must have the same value as the sum.
Decide which one the question wants.
$\text{answer} \geq 10$
The number of hot dogs must be a multiple of 10 and the number of buns a multiple of 8, and they must be equal: a common multiple.
List the multiples of 10 and test each with 8.
$10, \; 20, \; 30, \; 40$
8 does not divide 10, 20 or 30, but $40 \div 8 = 5$.
Read the answer and count the packs.
Match each pair of numbers to its greatest common factor.
| 3 | 7 | 4 | |
|---|---|---|---|
| 9 and 12 | |||
| 14 and 35 | |||
| 16 and 20 |
The greatest common factor of 42 and 54 is 6. Use it to find their least common multiple.
Divide the first number by the greatest common factor.
$42 \div 6 =$ m
This is the part of 42 that the other number does not share.
Divide the second number by the greatest common factor.
$54 \div 6 =$ n
This is the part of 54 that the first number does not share.
Multiply the greatest common factor by both quotients.
$\text{LCM} = 6 \times (\text{first quotient}) \times (\text{second quotient}) =$ lcm
The shared part is counted once, and each number's own part is added in, so both numbers divide the result.
Check by dividing the LCM by the first number.
$\text{LCM} \div 42 =$ back
A whole-number answer shows that 42 really divides the LCM.
Find the greatest common factor and the least common multiple of 14 and 35.
GCF = g and LCM = l
Use the greatest common factor to write 27 + 72 as a whole number times a sum of two whole numbers that share no factor except 1.
27 + 72 = f × (p + q)
Aisha waters the class tomato plants every 8 days and adds plant food every 10 days. Today Aisha did both. In how many days will both jobs fall on the same day again?
Both jobs fall on the same day again in answer days.
A coach packs snack kits for a team trip. She has 26 apples and 65 juice boxes. Every kit must hold the same number of apples and the same number of juice boxes, with nothing left over. What is the greatest number of kits she can make?
She can make kits kits, each with apples apples and other juice boxes.
In a toy robot, a small gear with 30 teeth turns a large gear with 36 teeth. A dot of paint marks one tooth on each gear, and the two dots touch now. How many full turns does the small gear make before the dots touch again?
The dots meet again after teeth teeth have passed, which is turns full turns of the small gear.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A coach packs snack kits for a team trip. She has 45 apples and 81 stickers. Every kit must hold the same number of apples and the same number of stickers, with nothing left over. What is the greatest number of kits she can make?
She can make kits kits, each with apples apples and other stickers.
You can find a GCF and an LCM and tell which a problem needs. Without looking: two sprinklers restart every 6 and 10 minutes. Do you want the GCF or the LCM, and when do they restart together?
17. Your turn: hot dogs come in packs of 10 and buns in packs of 8. What is the fewest hot dogs you can buy so that every hot dog gets a bun, with none of either left over?, step 3
$40 \text{ hot dogs}: \; 40 \div 10 = 4 \text{ packs}, \quad 40 \div 8 = 5 \text{ packs}$
40 is the LCM, so buy 4 packs of hot dogs and 5 packs of buns.