Imagine holding a completed Lego spaceship in your hands. If you drop it on the floor, it breaks down into smaller chunks of bricks. You can break those chunks down again and again until you are left with individual, unbroken Lego pieces that cannot be taken apart any further. In mathematics, numbers work the exact same way!
Every composite number can be broken down into individual prime numbers that act like unbreakable building blocks. The easiest and most visual way to find these building blocks is by drawing a factor tree. By checking factorizations on Calcsolver or splitting numbers into tree branches, middle school students can break down big numbers into prime factors with complete ease.
Mastering factor trees makes finding common denominators, simplifying square roots, and solving algebra fractions fun and simple.
What is Prime Factorization and How Does a Factor Tree Work
A prime number is a whole number greater than 1 that has only two factors: 1 and itself (such as 2, 3, 5, 7, 11, and 13). A composite number is any number that can be divided evenly by numbers other than 1 and itself (such as 12, 18, and 24).
Under the Fundamental Theorem of Arithmetic, every composite number can be factored into a unique product of prime numbers, like a unique DNA fingerprint.
A factor tree is a visual branching diagram used to break a composite number down into its prime factors. You start with the target number at the top (the trunk) and draw two branches down to any two numbers that multiply to make it. If a branch lands on a composite number, you split it again. When a branch lands on a prime number, you circle it: that branch is finished!
Also read: Modulo Operator in Math and Programming Clock Arithmetic and Remainders
You may also like: Binomial Expansion Formula Using Pascals Triangle to Expand Powers
Does It Matter Which Starting Numbers You Pick
One of the most comforting secrets about factor trees is that your starting choice does not matter at all! You will always end up with the exact same prime numbers at the bottom of the tree.
For example, if you factor the number 36: you could start with 6 times 6, or 4 times 9, or 2 times 18. Whichever path you choose, when all branches finish splitting, the circled prime numbers will always be: 2, 2, 3, and 3.
Step by Step Worked Factor Tree Example for Factoring 72
Let us build a complete factor tree for the number 72 step by step.
Step one: Write 72 at the top. Pick any two whole numbers that multiply to give 72. Let us choose 8 and 9. Draw two branches down to 8 and 9.
Step two: Look at the left branch (8). 8 is composite. Split 8 into 2 times 4. 2 is prime, so circle the 2. 4 is composite, so split 4 into 2 times 2. Both 2s are prime, so circle them.
Step three: Look at the right branch (9). 9 is composite. Split 9 into 3 times 3. Both 3s are prime, so circle both 3s.
Step four: Inspect all branch tips. All ends have circled prime numbers: three 2s and two 3s.
Step five: Write out the prime factorization: 72 = 2 x 2 x 2 x 3 x 3.
Step six: Write in exponential notation by combining identical factors: 72 = (2^3) x (3^2).
| Target Composite Number | Branch Splitting Path | Expanded Prime Factors | Compact Exponential Form |
|---|---|---|---|
| 12 | Split to 3 x 4 -> 3 x (2 x 2) | 2 x 2 x 3 | (2^2) x 3 |
| 24 | Split to 4 x 6 -> (2 x 2) x (2 x 3) | 2 x 2 x 2 x 3 | (2^3) x 3 |
| 60 | Split to 6 x 10 -> (2 x 3) x (2 x 5) | 2 x 2 x 3 x 5 | (2^2) x 3 x 5 |
| 72 | Split to 8 x 9 -> (2 x 2 x 2) x (3 x 3) | 2 x 2 x 2 x 3 x 3 | (2^3) x (3^2) |
| 100 | Split to 10 x 10 -> (2 x 5) x (2 x 5) | 2 x 2 x 5 x 5 | (2^2) x (5^2) |
The Division Ladder Method as an Alternative to Trees
While factor trees are visual and intuitive, some students prefer the division ladder method (upside-down division). You write your number inside an L-bracket and divide only by small prime numbers (2, 3, 5...) until the bottom quotient reaches 1.
For 60: divide by 2 to get 30; divide by 2 to get 15; divide by 3 to get 5; divide by 5 to get 1. The outside divisor column gives the exact same prime factors: 2 x 2 x 3 x 5.
Frequently Asked Questions About Factor Trees
Is the Number One Considered a Prime Factor
No. By mathematical definition, the number 1 is neither prime nor composite. It cannot be used as a branch on a factor tree because multiplying by 1 would create an infinite meaningless branch (e.g. 12 = 12 x 1 x 1 x 1).
How Do Factor Trees Help Simplify Square Roots
To simplify radicals like the square root of 72, look at its prime factors: (2 x 2) x 2 x (3 x 3). Every matching pair can leave the radical: 2 x 3 = 6 exits the radical, leaving the lone 2 inside: 6 times the square root of 2.
Can Any Number Have Two Different Prime Factorizations
No. The Fundamental Theorem of Arithmetic guarantees that the prime factorization of any positive integer is 100 percent unique, regardless of the order in which factors are written.