Question

Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish is correct.

Answer 1

Answer:

Trish is correct, since $4, 6, 8, 12$ are divisors of $48$.

Step-by-step explanation:

Trish is correct, since $4, 6, 8, 12$ are divisors of $48$. By factor decomposition we find the following product of prime numbers:

$48 = 2\times 2 \times 2 \times 2 \times 3$

$48 = 2^{4}\times 3$

The factor decomposition of $4, 6, 8, 12$ are, respectively:

$4 = 2^{2}$

$6 = 2\times 3$

$8 = 2^{3}$

$12 = 2^{2}\times 3$

Since each number is contained in the factor decomposition of 48, we can re-arrange $48$ in the following two ways:

Donna

$48 = (2\times 3) \times (2^{3})$

$48 = 6 \times 8$

Larry

$48 = 2^{2} \times (2^{2}\times 3)$

$48 = 4\times 12$