Question

darius created a factor tree and wrote the prime factorization of 60 shown here. factor tree: 60 (top) splits into 4 and 15; 4 splits into 2 (circled) and 2 (circled); 15 splits into 3 (circled) and 5 (circled); below: ( 60 = 2 \times 3 \times 5 ) what is darius’s error?

• darius should have started the factor tree with ( 2 \times 30 ).
• darius did not complete the factorization; 5 is a composite number.
• darius should have expressed 15 as ( 5 \times 5 ), not ( 3 \times 5 ).
• darius should have expressed 60 as ( 2^2 \times 3 \times 5 ), not ( 2 \times 3 \times 5 ).

Explanation:

1. Analyze each option:

• Option 1: Starting with \(2\times30\) is a valid start, but not an error. Any factor pair of 60 can start the tree.
• Option 2: 5 is a prime number, so this statement is wrong.
• Option 3: \(15 = 3\times5\) is correct (since \(5\times5 = 25

eq15\)), so this is wrong.

• Option 4: The factor 4 is decomposed into \(2\times2\), so the prime factors of 60 should include \(2^2\) (from 4) along with 3 and 5. Darius wrote \(2\times3\times5\) instead of \(2^2\times3\times5\), which is the error.

Answer:

Darius should have expressed 60 as \(2^{2}\times3\times5\), not \(2\times3\times5\).