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.
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Start by dividing by 2 (the smallest prime):
- \(10032 \div 2 = 5016\)
- \(5016 \div 2 = 2508\)
- \(2508 \div 2 = 1254\)
- \(1254 \div 2 = 627\)
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Now, 627 is not divisible by 2, so we try the next prime number, which is 3:
- \(627 \div 3 = 209\)
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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.)