To rewrite the number \(10032\) as a product of identical factors, we first need to factor it.
Let's start by factoring \(10032\):
- Check for divisibility by \(2\): \(10032\) is even, so we can divide by \(2\): \[ 10032 \div 2 = 5016 \]
- Continue factoring \(5016\): \[ 5016 \div 2 = 2508 \]
- Continue factoring \(2508\): \[ 2508 \div 2 = 1254 \]
- Continue factoring \(1254\): \[ 1254 \div 2 = 627 \]
- Now, since \(627\) is not even, check for divisibility by \(3\) (sum of digits \(6 + 2 + 7 = 15\) which is divisible by \(3\)): \[ 627 \div 3 = 209 \]
- Check \(209\) for divisibility (by \(11\)): \[ 209 = 11 \times 19 \]
Now we have the complete factorization of \(10032\): \[ 10032 = 2^4 \times 3^1 \times 11^1 \times 19^1 \]
To express \(10032\) as a product of identical factors, we can look for the greatest number of identical factors:
The highest power among the prime factors is \(2^4\). We can express it as: \[ 10032 = (2^2 \times 3 \times 11 \times 19)^2 \]
Calculating \(2^2 \times 3 \times 11 \times 19\): \[ 2^2 = 4 \] \[ 4 \times 3 = 12 \] \[ 12 \times 11 = 132 \] \[ 132 \times 19 = 2508 \]
Thus, we can write: \[ 10032 = (2508)^2 \]
So as a product of identical factors: \[ 10032 = (4 \times 3 \times 11 \times 19)^2 \] Or simply: \[ 10032 = (2508)^2 \]
Hence, the final answer is: \[ \boxed{(2508)^2} \]
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