To rewrite 10032 as a product of identical factors, we first need to factor it.
- Start by dividing by 2 (since 10032 is even): \[ 10032 \div 2 = 5016 \]
- Divide by 2 again: \[ 5016 \div 2 = 2508 \]
- Divide by 2 again: \[ 2508 \div 2 = 1254 \]
- Divide by 2 again: \[ 1254 \div 2 = 627 \]
- 627 is not even, so try dividing by 3: \[ 627 \div 3 = 209 \]
- 209 is not divisible by 3, but let's check if it's a prime. Checking 209:
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits is 11).
- Not divisible by 5 (doesn't end in 0 or 5).
- Dividing 209 by 7 gives approximately 29.857 (not an integer).
- Dividing 209 by 11 gives exactly 19.
Thus, we can factor 10032 as: \[ 10032 = 2^4 \times 3 \times 11 \times 19 \]
This expression indicates that we cannot fully express 10032 as a product of identical factors such as \( n^k \) where \( n \) is an integer (except for partial components).
However, we can express it using powers of its prime factors. The factorization indicates that it has no duplicate groups other than 2 raised to a power.
In conclusion, the prime factorization of 10032 is: \[ 10032 = 2^4 \times 3^1 \times 11^1 \times 19^1 \]
There isn't a single group of identical factors as you might have been looking for with just multiples (like \( n^2, n^3, \) etc.), but the closest simplified product of its prime factors is: \[ 10032 = 2^4 \cdot 3^1 \cdot 11^1 \cdot 19^1 \]
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