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

Answer 1

To express \( 10032 \) as a product of identical factors, we need to factor the number.

First, let's find the prime factorization of \( 10032 \):

1. Check for divisibility by \( 2 \): \[ 10032 \div 2 = 5016 \]

2. Continue dividing by \( 2 \): \[ 5016 \div 2 = 2508 \] \[ 2508 \div 2 = 1254 \] \[ 1254 \div 2 = 627 \] At this point, \( 627 \) is not divisible by \( 2 \).

3. Check for divisibility by \( 3 \): \[ 627 \div 3 = 209 \]

4. Next, check \( 209 \): \( 209 \) is not divisible by \( 3 \) or \( 5 \), but it is divisible by \( 11 \): \[ 209 \div 11 = 19 \] And \( 19 \) is a prime number.

So, the complete 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, we can rewrite it in terms of powers. The highest power for the group of identical factors can be chosen from the prime factors. This is usually represented in the form of a product of squares or cubes if it is possible.

Since we can factor \( 10032 \) more simply for identification:

1. We can group it as follows: \[ 10032 = (2^2 \times 3 \times 11 \times 19) \times (2^2) \] This indicates two identical factors, \( 10032 = 252 \times 40 \).

In summary, the product of identical factors is: \[ 10032 = 2^4 \times 3^1 \times 11^1 \times 19^1 \]

Thus, in a simplified form: \[ 10032 = (2^2 \times 3 \times 11 \times 19) \times (2^2) \]

Calculating \( 2^2 \): \[ 4 \times 3 \times 11 \times 19 \rightarrow 10032 \]

The answer can be returned succinctly: \[ 10032 = (252)^2 \]

So: \[ \boxed{252^2} \]

Answer 2

Rewrite the expression as a product of identical factors to simplify 12523 . Submit your answer in simplified form.

Answer 3

To express \( 12523 \) as a product of identical factors, we first need to determine its prime factorization.

Let's start by checking for divisibility by small prime numbers.

1. Divisibility by 2: \( 12523 \) is odd, so it is not divisible by \( 2 \).
2. Divisibility by 3: The sum of the digits \( 1 + 2 + 5 + 2 + 3 = 13 \) is not divisible by \( 3 \), so \( 12523 \) is not divisible by \( 3 \).
3. Divisibility by 5: The last digit is \( 3 \), so it is not divisible by \( 5 \).
4. Divisibility by 7: \[ 12523 \div 7 \approx 1789 \quad \text{(continue to check if } 1789 \text{ is prime)} \]
5. Divisibility of 1789: Continuing the factorization process, check smaller primes up to \( \sqrt{1789} \):
   • Not divisible by \( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \).
6. Divisibility by 31: \[ 1789 \div 31 = 57.096 \quad \text{(not an integer)} \]
7. Checking by larger primes: After testing various primes, we find:
   • \( 12523 = 7 \times 1789 \)
   • Continuing with \( 1789 = 13 \times 137 \)
   • And \( 137 \) is also a prime.

So the full prime factorization of \( 12523 \) is: \[ 12523 = 7 \times 13 \times 137. \]

This prime factorization cannot be expressed as a product of identical factors since none of the factors can be grouped into pairs that equal the number. It is primarily product of distinct prime numbers.

In conclusion, \( 12523 \) doesn't have any identical factors as it is composed of distinct primes. Therefore, while we cannot rewrite \( 12523 \) as a product of identical factors, we present the prime factorization: \[ \boxed{7 \times 13 \times 137}. \]

If you specifically need it in a product format other than distinct primes, please clarify the requirement for identical factors.