Question

Rewrite the expression as a product of identical factors to find the product for 10032
. Submit your answer in simplified form.



(1 point)

Answer 1

To rewrite the expression for 10032 as a product of identical factors, we need to factor it completely.

First, let's find the prime factorization of 10032.

1. Start by dividing by 2 (the smallest prime):
- \(10032 \div 2 = 5016\)
- \(5016 \div 2 = 2508\)
- \(2508 \div 2 = 1254\)
- \(1254 \div 2 = 627\)

2. Now, 627 is not divisible by 2, so we try the next prime number, which is 3:
- \(627 \div 3 = 209\)

3. Next, test 209 for divisibility by the next primes. It is not divisible by 3, 5, or 7, but:
- \(209 = 11 \times 19\) (since 11 and 19 are both primes)

Putting it all together:
- The prime factorization of 10032 is:
\[
10032 = 2^4 \times 3^1 \times 11^1 \times 19^1
\]

To express 10032 as a product of identical factors (if possible), we note that 2 is the only prime factor with a power that is greater than 1. Since \(2\) occurs \(4\) times in the factorization, we can also include the product of the other factors.

However, we can express 10032 as a product:
\[
10032 = (2^2 \times 3 \times 11 \times 19) \times (2^2) = 84 \times 2^4
\]

Thus, breaking this down into the largest identical product is limited, as identical repetitions need the factors to be the same. The best we can say for factorization into identical factors:
\[
10032 = 2^4 \cdot (3 \cdot 11 \cdot 19)
\]

In this case, the necessary product of two identical factors that equals \(10032\) doesn’t easily exist, but we transitively see that \(2^4\) = 16 and \(84\) can be kept in focus.

The final answer in simplified factored form would be:
\[
10032 = 2^4 \times 3 \times 11 \times 19 = 84 \times 16
\]

Thus, in request for a product of identical factors, it most aligns with the usable lower systematic calculations, given that \(10032\) does not reduce more identically into mere repetitions.

If you're looking for \(10032\) as a product:
\[
\text{Product: } 10032
\]
This reduces to an expression ideally as noted: \( \text{10032 = } (2^2 \cdot 84) = 10032; 4 \text{ is the maximum repetition here, effectively.} \)
(A caveat of the factor product lanes derived.)