Number theory, or the study of integers (the counting numbers 1, 2, 3…, their opposites –1, –2, –3…, and 0), has fascinated mathematicians for years. Prime numbers, a concept introduced to most students in Grades 4 and up, are fundamental to number theory. They form the basic building blocks for all integers.
A prime number is a positive integer (often called a natural number or counting number) that only has two factors, itself and one. Natural numbers that have more than two factors (such as 6, whose factors are 1, 2, 3, and 6), are said to be composite numbers. The number 1 only has one factor and isn’t usually considered prime or composite.
This article is broadly broken into two lessons, one that introduces the concept of prime factors and another that develops it. Both of them support the following standard:
- Key standard: Determine whether a given number is prime or composite, and find all factors for a whole number. (Grade 4)
Why do prime factors matter?
It’s the age-old question that math teachers everywhere contend with: When will I use this?
One notable real-world application of prime numbers is in cryptography, or the study of creating and deciphering codes. It is relatively easy to multiply two prime numbers. However, it can be extremely difficult to go the other direction and factor a number that has already been multiplied. To see what I mean, you can no doubt multiply 103,969 and 824,609 by hand even if it might take you a few minutes. But if you were tasked with figuring out the factors of 85,733,773,121 by hand, well . . . good luck.
Because of this feature about numbers, when a website sends and receives information securely—something important across the internet, but especially so for financial or medical websites, for example—you can bet there are prime numbers behind the scenes encrypting the information being sent, helping to ensure that anyone who tries to view it wouldn’t be able to make sense of it. Prime numbers also show up in a variety of surprising contexts, including physics, music, and even in the arrival of cicadas!
There is another place where prime numbers show up often, and it’s easy to overlook when discussing applications: math. The study of pure mathematics is a topic that people practice, study, and share without worrying about where else it might apply, similar to how a musician freely practices music without needing to ask how it applies elsewhere. Number theory is a deep math topic that is central to plenty of research papers, university courses, and other branches of mathematics. Mathematicians of all stripes no doubt encounter number theory many times along their academic and professional journeys.
Writing a product of prime factors
When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 72 can be written as a product of prime factors: \(72=2^3 \cdot 3^2\). The expression \(2^3 \cdot 3^2\) is said to be the prime factorization of 72.
The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely into a product of prime factors. Note that the same factors in a different order are considered identical. For example, \(3^2 \cdot 2^3\) is the same prime factorization as the one above, just not written in order from least to greatest prime factor. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations will be essentially the same.
Examine the two factor trees for 72 shown below.
When we get done factoring using either set of factors to start with, we still have three factors of 2 and two factors of 3, or \(2^3 \cdot 3^2\). This fundamental feature of 72 would remain true if we had instead factored 72 as 24 times 3, 4 times 18, or any other pair of factors for 72.
Knowing rules for divisibility is helpful when factoring a number. For example, if a whole number ends in 0, 2, 4, 6, or 8, we could always start the factoring process by dividing by 2. This calls attention to a special feature of the number 2. Because it only has two factors, 1 and 2, it is the only even prime number.
Another way to factor a number other than using factor trees is to divide the starting number by a prime number, then divide that quotient by a prime number, then continue dividing by prime numbers until there are no more prime numbers to divide. Here, 72 is written at the bottom, which each row above it representing another prime factor being factored out.
Once again, by multiplying all of the factors along the left side, we can see that \(72=2^3 \cdot 3^2\).
Critical to writing the prime factorization of a number is an understanding of exponents. An exponent indicates how many times a base is used as a factor. In the factorizations above, it corresponds to how many times each prime factor is repeated. For \(72=2^3 \cdot 3^2\), the 2 is used as a factor three times and the 3 is used as a factor twice.
There is a useful strategy that may help when trying to figure out whether a number is prime. Find the square root (with the help of a calculator if needed), and only check prime numbers less than or equal to it. For example, to see if 131 is prime, because the square root is between 11 and 12, we only need to check for divisibility by 2, 3, 5, 7, and 11. There is no need to check 13, since 132 = 169, which is greater than 131. If a prime number greater than 13 divided 131, then the other factor would have to be less than 13—which we’re already checking!
Lesson 1 – Introducing the concept: Finding prime factors
An important of practicing math is attending to precision. Make sure your students’ work is neat and orderly to help prevent factors getting mixed up or lost when constructing factor trees. Have them check their prime factorizations by multiplying the factors to see if they get the original number.
Prerequisite skills and concepts: Students will need to know and be able to use exponents. They will find it helpful to know the rules of divisibility for 2, 3, 4, 5, 9 and 10.
Write the number 48 on the board.
- Ask: Who can give me two numbers whose product is 48?
Students should identify pairs of numbers like 6 and 8, 4 and 12, or 3 and 16. Take one of the pairs of factors and create a factor tree for the prime factorization of 48 where all students can see it.
- Ask: How many factors of two are there? (4) How do I express that using an exponent?
Students should say to write it as \(2^4\). If they don’t, remind them that the exponent tells how many times the base is taken as a factor. Finish writing the prime factorization on the board as \(2^4 \cdot 3\). Next, find the prime factorization for 48 using a different set of factors.
- Ask: What do you notice about the prime factorization of 48 for this set of factors?
Students should notice that the prime factorization of 48 is \(2^4 \cdot 3\) for both of them.
- Say: There is a theorem in mathematics that says when we factor a number into a product of prime numbers, it can only be done one way, not counting the order of the factors.
Illustrate this concept by showing them that the prime factorization of 48 could also be written as \(3 \cdot 2^4\), but mathematically, that’s the same thing as \(2^4 \cdot 3\).
- Say: Now try one on your own. Find the prime factorization of 60 by creating a factor tree for 60.
Have all students independently factor 60. As they complete their factorizations, observe what students do and take note of different approaches and visual representations. Ask for a student volunteer to factor 60 for the entire class.
- Ask: Who factored 60 differently?
Have students who factored 60 differently (either by starting with different factors or by visually representing the factor tree differently) show their work to the class. Ask students to describe similarities and differences in the factorizations. If no one used different factors, show the class a factorization that starts with a different set of factors for 60 and have students identify similarities and differences between your factor tree and other students’.
- Ask: If I said the prime factorization of 36 is 22 • 9, would I be right?
The students should say no, because 9 is not a prime number. If they don’t, remind them that the prime factorization of a number means all the factors must be prime and 9 is not a prime number.
For additional practice, place the following numbers on the board and ask students to write the prime factorization for each one using factor trees: 24, 56, 63, and 46.
Lesson 2 – Developing the concept: Product of prime numbers
Now that students can find the prime factorization for numbers which are familiar products, it is time for them to use their rules for divisibility and other notions to find the prime factorization of unfamiliar numbers. Write the number 91 on the board.
- Say: Yesterday, we wrote some numbers in their prime factorization form.
- Ask: Who can write 91 as a product of prime numbers?
Students might say it can’t be done, because they will recognize that 2, 3, 4, 5, 9 and 10 don’t divide it. Some might think that 91 is prime. They may not try to see if 7 divides it, which it does. If they don’t recognize that 7 divides 91, demonstrate it for them. The prime factorization of 91 is \(7 \cdot 13\). Next, write the number 240 on the board.
- Ask: Who can tell me two numbers whose product is 240?
Students may name 10 and 24. If not, ask them to use their rules for divisibility to see if they can find two numbers. Create a factor tree for 240 like the one below.
- Ask: How many factors of two are there in the prime factorization of 240? (4) Who can tell me how to write the prime factorization of 240? (24 • 3 • 5)
Facilitate a discussion around different ways to factor 240 and the pros and cons of each method. If you start with 2 and 120, you end up with the same prime factorization in the end, but it looks like a “one-sided tree” that some students may find more difficult to work with. Have students identify ways that they prefer to factor and guide them to explain their reasoning.
- Say: Since the prime factorization of 240 is 24 • 3 • 5, the only prime numbers that divide 240 are 2, 3 and 5. Prime numbers like 7 and 11 will not divide the number, because they do not appear in the prime factorization of the number.
Call attention to the bases 2, 3, and 5 when making your point. The exponents don’t matter when discussing whether a different prime number will divide the original number. Write the number 180 on the board.
- Ask: What two numbers might we start with to find the prime factorization of 180? What other numbers could we use?
Encourage students to find a variety of pairs, such as 10 and 18 or 9 and 20. If no one mentions either pair, suggest them both as possibilities. Have half the students use 10 and 18 and the other half use 9 and 20. Have two students create the two factors for the class to see.
- Ask: If the prime factorization of a number is 22 • 5 • 7, what can you tell me about the number?
Some possible observations: Because 2 is a factor, the number is even. Because 2 and 5 are factors, 10 is a factor, so the number will end in a zero. Because 3 is not in the prime factorization, the number is not divisible by 3.
- Ask: If the prime factorization of a number is 33 • 11, what can you tell me about this number?
Repeat the previous exercise with a new number. Some possible observations: Because \(3^2\) is a factor, the number is divisible by 9 and the sum of the number’s digits is a multiple of nine. Because the product of odd numbers is always odd, the number is an odd number. They might also tell you that it is a composite number, five is not a factor of the number, and so on.
Give them the following numbers and ask them to find their prime factorization: 231, 117, and 175. Also give the following prime factorizations of numbers and ask them to write down at least two things they know about each of the numbers: \(3^2 \cdot 5^2\), \(2^3 \cdot 3 \cdot 13\), and \(2^2 \cdot 3 \cdot 5\). Adjust both the numbers and factorizations as needed to match what your students are ready for.
Wrap-up and assessment hints
Finding the prime factorization of numbers will strengthen your students’ basic fact fluency and understanding of multiplication. Students who do not know their basic multiplication facts will likely struggle with this, because they do not recognize products such as 24 or 63 readily. Turning the problem around and giving them the prime factorization of a number and asking them what they know about the number without multiplying it out is a good way to assess their understanding of the divisibility rules, the concept of factoring, and multiplication in general.
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This blog, originally published in 2021, has been updated for 2025.
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