Trish is correct because the prime factorization of a number is unique, regardless of how you choose to break the number down into factors. The prime factorization is determined by ultimately breaking down the number into its prime factors, which are the building blocks of whole numbers.
For the number 48, we can start with any pair of factors, whether it be 6 and 8, 4 and 12, or any other combination, and we will eventually arrive at the same set of prime factors.
Let’s illustrate this:
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Using Donna's factors (6 and 8):
- Start with 48 = 6 × 8
- Now factor 6: 6 = 2 × 3
- And factor 8: 8 = 2 × 4 (and then factor 4: 4 = 2 × 2)
- Therefore, we have:
- 48 = 6 × 8 = (2 × 3) × (2 × 4) = (2 × 3) × (2 × 2 × 2)
Now, combining all the prime factors:
- 48 = 2 × 2 × 2 × 2 × 3 = \(2^4 × 3^1\)
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Using Larry's factors (4 and 12):
- Start with 48 = 4 × 12
- Now factor 4: 4 = 2 × 2
- And factor 12: 12 = 3 × 4 (and 4 = 2 × 2)
- Therefore, we have:
- 48 = 4 × 12 = (2 × 2) × (3 × 4) = (2 × 2) × (3 × (2 × 2))
Combining all the prime factors again:
- 48 = 2 × 2 × 3 × 2 × 2 = \(2^4 × 3^1\)
In both methods, we ultimately arrive at the same prime factorization: \(2^4 × 3^1\). This demonstrates that it does not matter what initial factors you choose; the final result—the prime factorization—will always be the same as long as you correctly break down each factor into its prime components. Thus, Trish is correct in stating that the initial factors do not affect the final prime factorization of 48.