Question

Three students need to produce a prime factorization of

48. 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 because the prime factorization of 48 will be the same regardless of which factors are chosen as the initial factors in the tree.

To see why, let's look at the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

No matter which factors are chosen as the initial factors, the same prime factors will eventually be found.

For example, if Donna's method is used, we could start with 6 and 8:

6 = 2 × 3

8 = 2 × 2 × 2

Then we could continue to factor each of these numbers until we reach prime factors:

6 = 2 × 3

8 = 2 × 2 × 2

= 2 × 2 × 2 × 3

= 2³ × 3

Now we have found all of the prime factors of 6 and 8, and we can combine them to get the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

If Larry's method is used, we could start with 4 and 12:

4 = 2 × 2

12 = 2 × 2 × 3

Then we could continue to factor each of these numbers until we reach prime factors:

4 = 2 × 2

12 = 2 × 2 × 3

= 2² × 3

Now we have found all of the prime factors of 4 and 12, and we can combine them to get the prime factorization of 48:

48 = 2 × 2 × 2 × 2 × 3

As we can see, the same prime factors are found regardless of which factors are chosen as the initial factors in the tree. Therefore, Trish is correct that the initial factors of 48 do not affect the prime factorization.

Answer 2

students need to produce a prime factorization of 48. 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.Trish is correct because the initial factors of 48 do not affect the prime factorization. Prime factorization is the process of breaking down a composite number into its prime factors. The prime factors of a number are those factors that are prime numbers. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, which means that 48 can be expressed as a product of these prime factors. The order in which we choose the initial factors to start the prime factorization does not affect the result, as long as we continue to break down the resulting factors into their prime factors until we cannot break them down any further. Therefore, Trish is correct that the initial factors of 48 do not affect the prime factorization.