Question

Linda begins with the factors 4 and 18 to find the prime factorization of 72 . Will Linda find the same prime factorization for 72 as Li and Tim? Explain how you know.

Answer 1

Linda will indeed find the same prime factorization for 72 as Li and Tim because the prime factorization of a number is unique. By breaking down the factors 4 (2²) and 18 (2×9) into their prime components and combining them, she will arrive at the prime factorization of 72, which is 2³×3².

Step-by-step explanation:

Linda will find the same prime factorization for 72 as Li and Tim because the prime factorization of a number is unique, regardless of the factors you start with. Although Linda starts with the factors 4 (which is 2²) and 18 (which is 2×9), when these numbers are broken down into their prime factors, they will combine to form the prime factorization of 72.

The steps for finding the prime factorization are as follows:

1. Start with the given factors: 4 and 18.
2. Write down the prime factors of 4, which are 2×2.
3. Write down the prime factors of 18, which are 2×9, and 9 can be further factored into 3×3.
4. Combine the prime factors: 2×2 (from 4) and 2×9 (from 18), yielding 2³×3².

The prime factorization of 72 is 2³×3². This is a fundamental property of numbers called the Fundamental Theorem of Arithmetic, and it guarantees that prime factorization is unique.