Three students need to produce a prime factorization of 48. Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish is correct.
The correct answer and explanation is:
Correct Answer:
Trish is correct. The initial factors chosen in a factor tree do not affect the final prime factorization of a number.
Explanation (Approximately 300 Words):
Prime factorization is the process of breaking down a number into a product of prime numbers only. Prime numbers are numbers greater than 1 that have no other factors except 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).
Let’s consider the number 48.
Different people can start a factor tree in different ways:
- Donna starts with:
48 = 6 × 8
Then:
6 = 2 × 3
8 = 2 × 2 × 2
So, 48 = 2 × 3 × 2 × 2 × 2
Rearranged: 48 = 2⁴ × 3 - Larry starts with:
48 = 4 × 12
Then:
4 = 2 × 2
12 = 3 × 4 = 3 × 2 × 2
So, 48 = 2 × 2 × 3 × 2 × 2
Rearranged: 48 = 2⁴ × 3
Both Donna and Larry ended up with the same prime factors, even though they began with different pairs of factors.
Trish is absolutely correct because the order or method used to factor a number does not change the result of its prime factorization. No matter where you start, as long as you correctly break down the number into prime numbers, the final product will always be the same.
This is due to the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be written as a unique product of prime numbers, regardless of the order in which they are factored.
So, while different factor trees may look different, they all lead to the same prime factorization:
48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
✅ Conclusion: Trish is right—the starting point doesn’t matter; the prime factorization will always be the same.