Your child comes home with homework on prime and composite numbers. You remember the terms from school, but when they ask you to explain the difference, the words don't quite come together the way you'd like.
Here in Texas, prime and composite numbers show up in the TEKS as early as grade 4 and resurface in middle school when students start working with factors, multiples, and fractions. Getting comfortable with the concept now makes their second appearance a lot easier to manage.
Since we work with students at both of these stages regularly, we've put together a short refresher on prime and composite numbers, with examples, FAQs, and practice problems to help the topic land.
These two terms are easier to tell apart than they might seem.
The only thing we have to look for is how many numbers a given number can be divided by. In other words, we need to determine how many factors a number has.
A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, and 11.
A composite number is a whole number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, and 12.
A quick way to demonstrate this idea is to get a bunch of marbles and ask your student whether they can divvy up the marbles into equal rows.
If you give them 7 marbles, the only way to arrange them evenly is in one row of 7, or seven rows of 1. Since these are the only two options, we can determine that 7 is a prime number.
On the other hand, if you take 12 marbles and do the same exercise, you have several options: 2 rows of 6, 3 rows of 4, 4 rows of 3. That’s how you know that 12 is a composite number.
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The first thing to check is whether the number is even.
If the number is even and it's larger than 2, it's automatically composite.
This is because every even number larger than 2 can be divided by 2, which immediately gives it more than two factors. This is a very useful shortcut when looking for composite numbers.
If the number is odd, we want to look for factor pairs. A factor pair is simply two numbers that multiply together to give you the number you are testing.
For example, 3 and 6 are a factor pair of 18 because 3 × 6 = 18.
To find all factor pairs, start with 1 and work upward, testing each number to see whether it divides evenly. For 18, it would go like this:
1 × 18: yes
2 × 9: yes
3 × 6: yes
18 ÷ 4 = 4.5, so no
This might seem like a long and tedious process, but here’s the silver lining: you only need to keep testing until you reach the square root of the number you’re testing.
Why? Every factor below the square root automatically comes paired with a factor above it.
When you found that 3 goes into 18, you also found 6 at the same time, because 3 × 6 = 18. By the time you reach the square root, every pair has already been accounted for. For 18, the square root is roughly 4.2, so once you have tested up to 4, you have found everything there is to find.
If the only factor pair you found is 1 and the number itself, the number is prime. If any other pair turned up, it would be composite.
As students move into grades 6 and 7, prime and composite numbers become the foundation for prime factorization, which is how students find greatest common factors and least common multiples efficiently.
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Great job so far! Now let's put what we've learned to the test with a few more numbers.
Working through the factor pairs:
1 × 17: yes
2: 17 ÷ 2 = 8.5, not a whole number
3: 17 ÷ 3 = 5.67, not a whole number
4: 17 ÷ 4 = 4.25, not a whole number
5 × 5 = 25, which is bigger than 17, so we stop
The only factor pair is 1 and 17. 17 is therefore a prime number.
Counting up from one, we see:
1 × 24: yes
2 × 12: yes
3 × 8: yes
4 × 6: yes
5 × 5 = 25, which is bigger than 24, so we stop
We found the factors 1, 2, 3, 4, 6, 8, 12, and 24. The number is therefore a composite.
This one catches many students off guard. 49 is odd, so we can't use the even number shortcut. Working through the pairs:
1 × 49: yes
2: 49 ÷ 2 = 24.5, so no
3: 49 ÷ 3 = 16.33, so no
4: 49 ÷ 4 = 12.25, so no
5: 49 ÷ 5 = 9.8, so no
6: 49 ÷ 6 = 8.17, so no
7 × 7 = 49: yes
That single additional pair is enough. 49 is composite, with factors 1, 7, and 49.
This is one of the trickier numbers, as at a glance, it doesn’t “look” like a composite number. It therefore hammers home the importance of checking for factor pairs. It also puts into perspective the importance of knowing the times tables.
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Here are a few practice problems to check whether your student has understood the difference between prime and composite numbers. Check your answers at the bottom of the page!
Is 23 prime or composite? Show your work using the factor pairs method.
Is 36 prime or composite? List all factor pairs.
List all prime numbers between 20 and 40.
51 is prime because it doesn't divide evenly by 2 or 3. Is this correct? Explain.
Is 91 prime or composite?
Lastly, we want to tackle two questions that we get asked all the time when teaching prime and composite numbers. And that’s where 0 and 1 fit into the picture.
By definition, for a number to be a prime, it needs to have two factors: 1 and itself. The number 1 only has a single factor: itself. It is therefore neither a prime nor a composite number.
Neither. Every number multiplied by 0 equals 0, which means that 0 has an infinite number of factors. Both definitions require a finite, countable set of factors, so 0 doesn’t belong in either group.
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Mathnasium tutors have years of experience in identifying knowledge gaps and teaching math in a way that sticks.
Mathnasium is a math-only learning center dedicated to helping K-12 students learn and master math at every level, from the foundational concepts introduced in elementary school through the more advanced content of high school.
If your child could use some extra support with prime and composite numbers, or any of the topics that build on them, we're here to help.
Our approach, the Mathnasium Method™, is proprietary, personalized, and designed to make sure students understand the material rather than just getting through it.
Each student starts with a diagnostic assessment that tells us exactly where they are. From there, our tutors follow a personalized learning plan, working with students face-to-face in a caring and fun group environment.
Because we focus exclusively on math, everything we do is built around how students best absorb and retain mathematical concepts at every stage of the program.
And the results speak for themselves:
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93% of parents report an improved attitude toward math after attending Mathnasium
90% of students saw an improvement in their school grades
With over 1,100 centers, we bring the Mathnasium Method™ close to your community.
For families in and around Carrollton, TX, Mathnasium of Castle Hills brings that same trusted approach to the local community.
Here’s what one parent had to share about our center.
If you're looking to strengthen your child’s skills and build lasting confidence in math, schedule a free assessment at Mathnasium of Castle Hills and see real progress take shape, session by session.
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23 is prime. Pairs tested: 1 x 23, then 2 (23 ÷ 2 = 11.5), 3 (23 ÷ 3 = 7.67), 4 (23 ÷ 4 = 5.75). Since 5 x 5 = 25 > 23, we stop. The only factor pair is 1 and 23.
36 is composite. Factor pairs: 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6.
Prime numbers between 20 and 40: 23, 29, 31, 37.
51 is composite. 3 x 17 = 51, so its factors include 1, 3, 17, and 51.
91 is composite. 7 x 13 = 91, so its factors include 1, 7, 13, and 91.
Mathnasium of Castle Hills is a math-only learning center for K-12 students in Carrollton, TX. Trusted by over a million parents, Mathnasium uses personalized learning plans and the proprietary Mathnasium Method™ to help students catch up, keep up, and get ahead on their math journey.
Our specially trained tutors deliver face-to-face instruction in a supportive and fun small-group environment, working with students both in center and online to develop a deep understanding of math, build confidence, and improve academic performance.
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