Greatest common factor and least common multiple
Common factors as shared rectangles, common multiples as shared skip counts, and choosing GCF or LCM
About four lessons of 45 to 60 minutes
Packing boxes evenly and lights that blink together
You are making party bags from 12 granola bars and 18 juice boxes. Every bag has to be identical, with nothing left over. How many bags can you make? The answer is hiding in the numbers that divide both 12 and 18, and the biggest one that works, 6, is exactly the greatest common factor.
Now picture two string lights: the red one blinks every 4 seconds and the blue one every 6 seconds. They just flashed together. When will they next flash at the same instant? Count in 4s, count in 6s, and find the first time both land together. That first shared moment, 12 seconds, is the least common multiple. Today you will build both ideas from rectangles and number lines, then learn to tell which one a problem is asking for.
- 12 granola bars and 18 juice boxes into equal bags6 bags is the most you can make evenly, so GCF(12, 18) = 6
- A red light every 4 secondsit flashes at 4, 8, 12, 16, 20, the multiples of 4
- A blue light every 6 secondsit flashes at 6, 12, 18, 24, the multiples of 6
- Both flash together at 12 seconds12 is the first shared multiple, so LCM(4, 6) = 12
What students will be able to do
Students will find the greatest common factor of two numbers up to 100 by listing or rectangle-building their common factors, find the least common multiple of two numbers by listing multiples until one is shared, tell a GCF situation from an LCM situation, and connect the two ideas to sharing into equal groups and to events lining up.
- I can list the common factors of two numbers and pick out the greatest.
- I can list the common multiples of two numbers and pick out the least.
- I can build a shared rectangle to show a common factor.
- I can decide whether a word problem needs the GCF or the LCM.
- I can explain why factors run out but multiples never do.
Standards this unit teaches
- 6.NS.B.4Common Core (US)Greatest common factor and least common multiple
Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1 to 100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
- 4.OA.B.4Common Core (US)Factor pairs, multiples, primes and composites (Grade 4 foundation)
Find all factor pairs for a whole number in the range 1 to 100. Recognise that a whole number is a multiple of each of its factors. This Grade 4 standard is the factor and multiple groundwork that greatest common factor and least common multiple build directly on.
- AC9M6N01Australian Curriculum v9 (ACARA)Factors, multiples and divisibility (Year 6)
Express natural numbers as products of their factors, recognise multiples and decide whether one number divides another. Australia meets the factor and multiple structure this unit rests on at Year 6, though ACARA does not name greatest common factor and least common multiple as a separate Year 6 outcome, so this unit teaches a US-specific pairing that draws on the same Australian content.
- AC9M6N04Australian Curriculum v9 (ACARA)Add and subtract fractions with related denominators (Year 6)
Solve problems involving addition and subtraction of fractions with the same or related denominators. Finding a common denominator to add fractions is one of the everyday uses of the least common multiple, so this unit supports the Year 6 fraction work.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Factors, multiples and primesthe factor pairs and multiples this unit combines
- How to teach factorsa factor divides a number exactly, with nothing left over
- Prime and composite numbersthe building blocks that make listing factors quicker
- Multiplicationa factor pair is a multiplication fact, and multiples are its skip count
- Skip countingcounting in equal steps lists the multiples of a number
Words to teach and display
- Factor
- a whole number that divides another exactly, with no remainder
- Multiple
- the result of multiplying a number by a whole number, so 12 is a multiple of 4
- Common factor
- a number that is a factor of two numbers at once, such as 6 for 12 and 18
- Greatest common factor (GCF)
- the largest number that divides both numbers exactly
- Common multiple
- a number that is a multiple of two numbers at once, such as 12 for 4 and 6
- Least common multiple (LCM)
- the smallest number both numbers divide into
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Common factors are shared rectangles
ConcreteStart with the factor rectangles students already know. Give each pair 12 counters and then 18 counters, and ask them to make equal rows with each. A rectangle only works when the counters split into equal rows with none left over, so every rectangle reveals a factor. The interesting question today is a factor that works for both numbers at once, a common factor.
Both 12 and 18 can be built into rows of 6: 12 makes 2 rows of 6, and 18 makes 3 rows of 6. So 6 is a factor of both numbers, a common factor. It is also the largest number that works for both, which is why we call it the greatest common factor of 12 and 18.
Both numbers can also be built into rows of 3 (12 as 4 rows of 3, 18 as 6 rows of 3), so 3 is a common factor too. But 3 is not the greatest. When a problem asks you to share into the largest possible equal groups, it is asking for the greatest common factor.
- Show a rectangle that proves 3 is a factor of both 12 and 18.
- Why can 6 not be beaten as a common factor of 12 and 18?
- What does it mean for a number to be a common factor of two numbers?
2. Finding the greatest common factor
PictorialRectangles make the idea concrete, but for larger numbers we list. Write out all the factors of each number in order, ring the ones that appear in both lists, and take the largest. Listing in order, from 1 and the number inward, is what stops a factor being missed.
For 12 and 18: the factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18. The numbers in both lists are 1, 2, 3 and 6. The greatest of those is 6, so the greatest common factor of 12 and 18 is 6.
The greatest common factor is never bigger than the smaller of the two numbers, because a factor cannot exceed its own number. And it is always at least 1, because 1 divides every number. So the GCF sits somewhere between 1 and the smaller number.
Find the greatest common factor of 16 and 24.
- List the factors of 16: 1, 2, 4, 8, 16.
- List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Ring the factors in both lists: 1, 2, 4 and 8. The greatest is 8.
Answer: The greatest common factor of 16 and 24 is 8.
- List the common factors of 16 and 24. Which is the greatest?
- Why can the GCF of 16 and 24 not be larger than 16?
- What is the greatest common factor of any number and itself?
3. Common multiples on the number line
PictorialNow turn around and look up instead of down. The multiples of a number are its skip count: the multiples of 4 are 4, 8, 12, 16, 20, 24 and on forever. Draw them as equal jumps along a number line. A common multiple is a number that both skip counts land on, and the first one they share is the least common multiple.
Count in 4s and count in 6s along two lines to the same scale. The 4s land on 4, 8, 12, 16, 20, 24. The 6s land on 6, 12, 18, 24. Both lines hit 12 and both hit 24, so 12 and 24 are common multiples of 4 and 6. The first shared landing, 12, is the least common multiple.
Unlike factors, which run out, multiples never end, so there are infinitely many common multiples. That is why we want the least one: it is the smallest number both counts reach, and every other common multiple is a multiple of it (24, 36 and 48 are all multiples of 12).
- Which numbers do both the 4s count and the 6s count land on, up to 24?
- Why is 12 the least common multiple and not 24?
- List the first three multiples of 5 and the first three of 3. Do any match yet?
4. Finding the least common multiple
PictorialThe listing method works without a number line too. Write the multiples of each number in order and keep going until a number shows up in both lists. The first one to appear in both is the least common multiple. Keep both lists growing at the same time so you do not overshoot.
For 6 and 8: multiples of 6 are 6, 12, 18, 24, 30, and multiples of 8 are 8, 16, 24, 32. The first number in both lists is 24, so the least common multiple of 6 and 8 is 24.
A useful check: the least common multiple is never smaller than the larger of the two numbers, and never larger than the two multiplied together. When two numbers share no common factor other than 1, the LCM is exactly their product (LCM of 3 and 5 is 15). When they do share a factor, the LCM is smaller than the product, because the shared factor is not double counted.
Find the least common multiple of 6 and 8.
- List multiples of 6: 6, 12, 18, 24, 30, 36.
- List multiples of 8: 8, 16, 24, 32, 40.
- The first number that appears in both lists is 24.
Answer: The least common multiple of 6 and 8 is 24.
- Find the least common multiple of 4 and 10.
- Two numbers share no factor except 1. How do you find their LCM quickly?
- Why must the LCM of 6 and 8 be at least 8?
5. Which do you need, GCF or LCM?
AbstractThe hardest part is not the arithmetic, it is deciding which tool a problem calls for. Two questions settle it. Are you breaking amounts down into the largest equal groups? That is the greatest common factor. Are you waiting for separate cycles to line up, or building up to a shared total? That is the least common multiple.
Greatest common factor problems share or split: the most identical party bags from 12 bars and 18 juices, the largest square tile that paves a room exactly, simplifying a fraction to its lowest terms. You are looking downward for the biggest piece that fits both.
Least common multiple problems are about things repeating until they coincide: two lights blinking together, buses that leave every 4 and every 6 minutes leaving together again, or finding a common denominator to add two fractions. You are looking upward for the first shared point.
A quick self-check on the answer: a GCF is at most the smaller number, while an LCM is at least the larger number. If your answer is bigger than both starting numbers, you have found a multiple, so it should be an LCM question, and the other way around.
Two problems. (a) You have 12 granola bars and 18 juice boxes and want the most identical party bags with nothing left over. (b) A red light blinks every 4 seconds and a blue light every 6 seconds, and they just blinked together. When do they next blink together?
- (a) is a sharing-into-largest-groups problem, so find the greatest common factor of 12 and 18. Common factors are 1, 2, 3, 6, and the greatest is 6.
- So 6 bags is the most, each with 12 divided by 6 = 2 bars and 18 divided by 6 = 3 juice boxes.
- (b) is a cycles-lining-up problem, so find the least common multiple of 4 and 6. Multiples of 4 are 4, 8, 12, and of 6 are 6, 12, so the first shared is 12.
Answer: (a) 6 party bags, each with 2 bars and 3 juice boxes. (b) The lights next blink together after 12 seconds.
- A problem asks for the biggest tile that fits a floor exactly. GCF or LCM?
- A problem asks when two timetables next match. GCF or LCM?
- Your answer is larger than both starting numbers. Which did you probably need?
Common misconceptions and how to address them
MisconceptionGreatest common factor and least common multiple are swapped, so a student gives a big multiple when the GCF is wanted.
Why it happens: The two are opposite directions of the same factor and multiple relationship, and the long names blur together.
How to address it: Anchor the directions out loud: factors go down and stop, so the GCF is small; multiples go up and never end, so the LCM is large. A GCF is at most the smaller number, an LCM is at least the larger.
MisconceptionA common factor is taken that is not the greatest, such as choosing 3 instead of 6 for 12 and 18.
Why it happens: The first shared factor a student spots feels like the answer, so they stop before checking for a larger one.
How to address it: List every common factor and only then take the largest. For 12 and 18 the common factors are 1, 2, 3 and 6, so 6, not 3, is the greatest.
MisconceptionThe multiples lists are stopped too early, so the first shared multiple is missed.
Why it happens: A student writes only two or three multiples of each and gives up before the lists overlap.
How to address it: Keep both lists growing together until a number appears in both. For 6 and 8 you must reach 24 before they meet, so short lists would miss it.
MisconceptionThe least common multiple is always the two numbers multiplied together.
Why it happens: It is true when the numbers share no factor, so students overgeneralise it to every pair.
How to address it: Show that 4 times 6 is 24 but the LCM of 4 and 6 is 12, because 4 and 6 share the factor 2. Multiplying only gives the LCM when the numbers share no common factor except 1.
MisconceptionThe greatest common factor is thought to be always 1.
Why it happens: Students meet coprime examples like 8 and 15 first, where the only common factor is 1, and assume it always is.
How to address it: Contrast 8 and 15 (GCF 1) with 12 and 18 (GCF 6). Numbers that share factors have a GCF bigger than 1, so you must actually check the common factors.
Misconception1 is left out of the common factors, so a student says two numbers have no common factor.
Why it happens: 1 feels too trivial to count, but it divides every whole number.
How to address it: Remind students that 1 is a factor of everything, so every pair of numbers has at least the common factor 1. Numbers whose only common factor is 1 are called coprime, not factorless.
Guided practice (with answers)
1. List the common factors of 12 and 18, then give the GCF.
Answer: Common factors: 1, 2, 3, 6. The greatest common factor is 6.
2. Find the greatest common factor of 20 and 30.
Answer: 10. Factors of 20 are 1, 2, 4, 5, 10, 20 and of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The common ones are 1, 2, 5, 10, and 10 is greatest.
3. Find the least common multiple of 3 and 5.
Answer: 15. They share no factor except 1, so the LCM is just 3 times 5.
4. Find the least common multiple of 4 and 6.
4s reach 4, 8, 12 and 6s reach 6, 12. First shared is 12. Answer: 12. Multiples of 4 are 4, 8, 12 and of 6 are 6, 12, so 12 is the first shared.
5. You have 24 red and 36 blue beads to thread onto identical bracelets with none left over. What is the most bracelets you can make?
Answer: 12. That is GCF(24, 36), so each bracelet gets 2 red and 3 blue beads.
6. Buses leave the depot every 10 minutes and every 15 minutes and just left together. When do they next leave together?
Answer: After 30 minutes, the least common multiple of 10 and 15.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Begin with finding common factors and multiples, then move to the GCF-or-LCM word problems once both are secure.
Differentiation
- Keep counters or square tiles out so students build the shared rectangle before listing factors.
- Give a printed factor list or a multiplication grid so the search for factors is reading, not recall.
- Start with pairs that share obvious factors, such as 6 and 12 or 10 and 20, before awkward pairs.
- Use a two-column recording sheet, one column per number, and circle the shared entries.
- Introduce prime factorisation as a faster route: GCF multiplies the shared prime factors, LCM takes the highest power of each.
- Explore three numbers at once, such as the LCM of 4, 6 and 8.
- Investigate coprime pairs (GCF 1) and describe what they have in common.
- Connect the LCM to adding fractions with unlike denominators, such as 1/4 + 1/6 using a denominator of 12.
Assessment: exit ticket
A three-question exit ticket for the last five minutes. It samples finding the GCF, finding the LCM, and choosing the right one from a situation.
1. Find the greatest common factor of 18 and 24.
Answer: 6. Common factors are 1, 2, 3, 6, and 6 is the greatest.
2. Find the least common multiple of 6 and 9.
Answer: 18. Multiples of 6 are 6, 12, 18 and of 9 are 9, 18.
3. You want the largest square tiles that pave a 12 by 18 floor exactly. GCF or LCM, and what is it?
Answer: GCF, because you are finding the biggest equal piece that fits both. GCF(12, 18) = 6, so 6 by 6 tiles.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 common factors as rectangles (section 1), Lesson 2 finding the GCF (section 2), Lesson 3 common multiples and the LCM (sections 3 and 4), Lesson 4 choosing GCF or LCM (section 5) plus the exit ticket.
- Language to keep saying: factors go down and stop, multiples go up and never end, a GCF is at most the smaller number, an LCM is at least the larger. These four phrases pre-empt most of the misconceptions.
- Keep counters or tiles available through the pictorial sections. Building the shared rectangle is what makes a common factor concrete before it becomes a list.
- The number-line diagrams use a step of 2 so every multiple of 4 and 6 lands on a labelled tick and the two lines share a scale. Point out that 12 and 24 sit directly under each other on both lines.
- Curriculum note and a US and AU alignment: the US pairs greatest common factor and least common multiple together in Grade 6 (6.NS.B.4), also asking for the distributive property link such as 36 + 8 = 4 times (9 + 2). ACARA covers the underlying factors, multiples and divisibility at Year 6 (AC9M6N01) but does not isolate GCF and LCM as a named outcome, so treat this as a US-shaped unit resting on shared Australian content.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.