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

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

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* 2 * 2 * 2 * 3`

`48 = 2^(4)* 3`

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

`4 = 2^(2)`

`6 = 2* 3`

`8 = 2^(3)`

`12 = 2^(2)* 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* 3) * (2^(3))`

`48 = 6 * 8`

Larry

`48 = 2^(2) * (2^(2)* 3)`

`48 = 4* 12`