A factor is a number that divides another number exactly, and a multiple is the number you get when you multiply by a whole number. I find the greatest common factor by listing the factors of both numbers and picking the biggest shared one. I find the least common multiple by listing multiples and picking the smallest shared one.
What Is a Factor?
A factor is a whole number that divides another number exactly, with nothing left over, so 3 is a factor of 12 because 12 divided by 3 is 4 with no remainder. The full list of factors for a number is always short and fixed, never endless.
Think of sharing 12 pencils into equal piles. You can make piles of 1, 2, 3, 4, 6 or 12, and each pile size is a factor of 12. A pile of 5 will not work, because 5 does not divide 12 exactly.
I ask children to work in pairs from the ends: 1 pairs with 12, 2 pairs with 6, 3 pairs with 4. The pairs meet in the middle, and that’s how I know no factor is missed. The factor pair habit saves more marks than any trick I know.
What Is a Multiple?
A multiple is the number you get when you multiply a number by a whole number, so the multiples of 7 are 7, 14, 21, 28 and 35, and the list never ends. Each step adds the same number again.
Say the 7 times table out loud and you are listing multiples of 7. The table never stops, so a number has endless multiples, unlike its short list of factors. That one difference sorts out most confusion between the two words.
Listing multiples in a row makes a number pattern, and I show children how the same gap repeats at every step. I explain patterns in full in my Number Patterns guide, where the rule behind the list is the whole game.
How Do You Find the Greatest Common Factor (GCF)?
To find the greatest common factor, list all the factors of each number, circle the factors they share, and pick the biggest one. For 12 and 18, the shared factors are 1, 2, 3 and 6, so the greatest common factor is 6.
Work in factor pairs again, so nothing is missed: 12 gives 1 and 12, 2 and 6, 3 and 4. For 18, the pairs are 1 and 18, 2 and 9, 3 and 6. The shared factors stand out fast once both lists are full.
This is the skill Common Core standard 6.NS.B.4 asks for: find the greatest common factor of two whole numbers up to 100, and the least common multiple of two whole numbers up to 12. The standard keeps the numbers small on purpose, so the method, not the arithmetic, does the work.
How Do You Find the Least Common Multiple (LCM)?
To find the least common multiple, list the multiples of each number in order and pick the smallest multiple they share. For 4 and 6, the multiples of 4 are 4, 8, 12 and 16, the multiples of 6 are 6, 12 and 18, and the smallest shared multiple is 12.
Start both lists at the same time and stop at the first match. Children who list one number fully first often run too far and miss the small shared one. The first match is the answer, and there is no need to list further.
A quick check helps: the LCM is never smaller than the bigger number. If your answer for 4 and 6 came out as 4, something went wrong, because 4 is not a multiple of 6.
What Mistakes Do Children Make Most With Factors and Multiples?
The biggest mistake is mixing up the two words, because a factor divides into a number while a multiple is made by multiplying it. Children also stop the factor list too early, missing the last pair, and then the common factor they pick is wrong.
I fix the mix-up with one sentence: factors fit inside, multiples grow outside. Say it before every question, and the two words stop swapping places. For the early stop, I make children check the last pair first: does the number divide by itself? It always does.
One more trap: children list factors of 24 and stop at 6, forgetting 8, 12 and 24. Work the pairs to the middle, and the list checks itself.
A Real-Life Case: Tiling a Floor
Here is one case, worked in full, the way I do it in class.
Step 1: A floor is 24 feet long and 18 feet wide. I want the biggest square tiles that fit exactly, with no cutting and no gaps, so I need the greatest common factor of 24 and 18.
Step 2: List the factors. For 24: 1, 2, 3, 4, 6, 8, 12, 24. For 18: 1, 2, 3, 6, 9, 18. The shared factors are 1, 2, 3 and 6, so the biggest square tile is 6 feet by 6 feet.
Step 3: Check it: 24 divided by 6 is 4 tiles along the length, and 18 divided by 6 is 3 tiles across the width. That is 12 tiles in all, and not one tile needs cutting.
Try It Yourself: 3 Practice Questions
Cover the answers, try all three, then check.
- List all the factors of 24.
- List the first 5 multiples of 7.
- Find the LCM of 4 and 6.
Answers
- 1, 2, 3, 4, 6, 8, 12, 24. Work in pairs: 1 and 24, 2 and 12, 3 and 8, 4 and 6.
- 7, 14, 21, 28, 35. Each step adds 7.
- 12. Multiples of 4 are 4, 8, 12; multiples of 6 are 6, 12; the smallest shared multiple is 12.
Key Takeaways
- Factors fit inside: they divide the number exactly, and the list is short.
- Multiples grow outside: they go on for ever, in equal steps.
- GCF: list factors, circle the shared ones, pick the biggest.
- LCM: list multiples, pick the smallest shared one.
- Check the pairs: work factors in pairs so nothing is missed.
If your child answered all three questions, she is ready for a longer page. For more free practice at this level, browse our Grade 6 math worksheets page. Every sheet is free to print.
Frequently Asked Questions
What is the difference between a factor and a multiple?
A factor divides into a number exactly, like 3 into 12, while a multiple is made by multiplying, like 12, 24 and 36 are multiples of 12. Factors are short lists; multiples go on for ever.
Is the greatest common factor ever bigger than the numbers themselves?
No. A common factor must divide both numbers, so it can never be bigger than the smaller one. The greatest common factor of 12 and 18 is 6, which is smaller than both.
How do you find the LCM of two numbers quickly?
List the multiples of the bigger number first and check each one against the smaller number. The first multiple of the bigger number that the smaller number divides is the LCM.