Prime Factorization & Factor Trees: Fifth Grade Mathematics

Prime factorization decomposes any composite number into an exclusive product of prime numbers, known as the prime fingerprint of that number.

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every composite number can be written as a product of prime numbers in exactly one unique way, regardless of the order of the factors.

+--------------------------------------------------------------------------+
|                  PRIME FACTORIZATION USING FACTOR TREES                  |
|                                                                          |
|       Target Number: 60                                                  |
|                                                                          |
|               Tree A                          Tree B                     |
|                 60                              60                       |
|                /  \                           /  \                      |
|               6    10                        4    15                     |
|              / \   / \                      / \   / \                    |
|             (2)(3)(2)(5)                    (2)(2)(3)(5)                 |
|                                                                          |
|       Both trees yield: 2 * 2 * 3 * 5 = 2^2 * 3 * 5                     |
+--------------------------------------------------------------------------+

How to Build a Factor Tree

  1. Write the composite number at the top.
  2. Choose any two factors whose product equals that number and draw two downward branches.
  3. If a factor is prime, circle it. It cannot be branched further.
  4. If a factor is composite, branch it again into two factors.
  5. Continue branching until every leaf at the bottom is a circled prime number.
  6. Write the prime numbers in ascending order using multiplication symbols or exponents.

Writing Prime Factorizations with Exponents

When a prime factor appears more than once, use exponent notation to write it compactly. For 72:
- 72 = 2 * 2 * 2 * 3 * 3

- Using exponents: 2^3 * 3^2.


Division Ladder Method (Alternative to Factor Trees):
    2 | 72
    2 | 36
    2 | 18
    3 |  9
    3 |  3
         1   <--- Prime factors on the left: 2, 2, 2, 3, 3 = 2^3 * 3^2

Exercises and Solutions

Exercise 1: Constructing a Factor Tree. Find the prime factorization of 84 using a factor tree. Write the result in expanded prime form and exponential form.

Solution 1: Start with 84 at the top. Branch 84 into 4 and 21. Branch 4 into 2 * 2 (both 2s are prime, so circle them). Branch 21 into 3 * 7 (both 3 and 7 are prime, so circle them). The circled prime factors are 2, 2, 3, and 7. In expanded prime form: 2 * 2 * 3 * 7. In exponential form: 2^2 * 3 * 7. (Verification: 4 * 3 * 7 = 12 * 7 = 84).

Exercise 2: Comparing Different Starting Factors. Demonstrate that starting a factor tree for 90 with 9 * 10 produces the exact same prime factorization as starting with 2 * 45.

Solution 2: For the first tree, start with 9 * 10: 9 branches into 3 * 3 (both prime), and 10 branches into 2 * 5 (both prime). The prime factors are 3, 3, 2, and 5, which arranged in ascending order is 2 * 3^2 * 5. For the second tree, start with 2 * 45: 2 is prime (circle it); 45 branches into 5 * 9 (5 is prime); 9 branches into 3 * 3 (both prime). The prime factors are 2, 5, 3, and 3, which arranged in ascending order is 2 * 3^2 * 5. Both trees produce the identical prime factorization, confirming the Fundamental Theorem of Arithmetic.