Question

georgette created a factor tree and wrote the prime factorization of 72 shown here.

what is georgettes error?

• georgette should have started the factor tree with \\(2 \times 36\\).
• georgette did not complete the factorization; 9 is a composite number.
• georgette should have expressed 4 as \\(4 \times 1\\), not \\(2 \times 2\\).
• georgette should have expressed 8 as \\(4 \times 4\\), not \\(2 \times 4\\).

Explanation:

Analyze the factor tree

The factor tree starts with \(72\) split into \(8\) and \(9\). The branch for \(8\) is correctly factored down to \(2 \times 4\), and then \(4\) is factored to \(2 \times 2\). This yields three prime factors of \(2\), which is written as \(2^3\).

Identify the uncompleted branch

The branch ending in \(9\) is circled as if it is a prime factor. However, \(9\) is a Composite Numbers because it has factors other than \(1\) and itself. Specifically, \(9 = 3 \times 3\).

Determine the error

Since \(9\) is not a Prime Numbers, the factorization is incomplete. Georgette should have factored \(9\) into \(3 \times 3\) to find the correct prime factorization: \(72 = 2^3 \times 3^2\).

Match with the options

The second option states: "Georgette did not complete the factorization; 9 is a composite number." This perfectly describes the error.

Answer:

• Georgette should have started the factor tree with 2 × 36.
• Georgette did not complete the factorization; 9 is a composite number. (Correct answer)
• Georgette should have expressed 4 as 4 × 1, not 2 × 2.
• Georgette should have expressed 8 as 4 × 4, not 2 × 4.