A factor tree is a visual representation that shows how a particular number can be expressed as a product of its prime factors. For the number 60, we can generate two distinct factor trees.
Factor Tree 1
1. Start with the number 60 at the top.
2. The first step is to divide 60 by its smallest prime factor, which is 2:
- 60 ÷ 2 = 30
3. Now, we can further break down 30:
- 30 ÷ 2 = 15
4. Next, we take the number 15 and divide it by its smallest prime factor, which is 3:
- 15 ÷ 3 = 5
5. Since 5 is also a prime number, we stop here.
So, the prime factorization for this tree is:
- 60 = 2 × 2 × 3 × 5
Factor Tree 2
1. Again, we start with 60 at the top.
2. This time, let’s divide 60 by a different factor, which is 3:
- 60 ÷ 3 = 20
3. Now, we break down 20:
- 20 ÷ 2 = 10
4. Then, we can further break down 10:
- 10 ÷ 2 = 5
5. As before, we stop here since 5 is a prime number.
Thus, the prime factorization for this second tree is:
- 60 = 3 × 2 × 2 × 5
Final Thoughts
Despite using different routes to arrive at the prime factorization, both factor trees articulate the same fundamental decomposition of the number 60:
- 60 = 2 × 2 × 3 × 5
- 60 = 3 × 2 × 2 × 5
These trees illustrate the flexibility in factorization while emphasizing that mathematical relationships can often be expressed in multiple ways.