To express 10032 as a product of identical factors, we need to factor the number.
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We start with the prime factorization of 10032:
- 10032 is even, so we can divide by 2: \( 10032 ÷ 2 = 5016 \)
- 5016 is also even: \( 5016 ÷ 2 = 2508 \)
- 2508 is still even: \( 2508 ÷ 2 = 1254 \)
- 1254 is even: \( 1254 ÷ 2 = 627 \)
Now we have \( 10032 = 2^4 × 627 \).
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Next, we factor 627. It is not even, so we check for divisibility by 3:
- The sum of the digits \( 6 + 2 + 7 = 15 \) is divisible by 3. \( 627 ÷ 3 = 209 \)
Therefore, we can update our factorization: \( 10032 = 2^4 × 3 × 209 \).
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Now, we check if 209 can be factored further. It is not even and:
- The sum of the digits \( 2 + 0 + 9 = 11 \) is not divisible by 3.
- It does not end in 0 or 5, so it is not divisible by 5.
- Checking for divisibility by 7, we find \( 209 ÷ 7 \approx 29.857 \), so it's not divisible.
- Checking by 11 gives \( 209 ÷ 11 = 19 \), and both 11 and 19 are primes.
Putting it all together: \( 10032 = 2^4 × 3 × 11 × 19 \).
We have the prime factorization of 10032 as \( 10032 = 16 × 3 × 11 × 19 \).
To rewrite this as a product of identical factors:
- Notice that \( 10032 = (2^2 × 3 × 11 × 19) × 2^2 = 84 × 12 \), or we could say
- \( 10032 = (2^2) \times (3 \times 11 \times 19) \)
Thus, express 10032 as: \( 10032 = (2^2 \times 3 \times 11 \times 19) = (4 × 3 × 11 × 19) = (12 × 11 × 19) = (132 × 19) \)
So, the simplest product of identical factors is: \( 10032 = 2^4 × 3 × 11 × 19 \).
If asked further, one possible simplified product as a square could be expressed as \( (2^2 × \sqrt{3 × 11 × 19})^2 = (4 × \sqrt{627})^2 \), but does not yield identical integers.
Final Answer: The simplified factorization of 10032 is:
\[ 10032 = 2^4 × 3 × 11 × 19 \].