Question

Rewrite the expression as a product of identical factors to find the product for 10032
.

Answer 1

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

Let's start by factoring \(10032\):

1. **Check for divisibility by \(2\)**:
\(10032\) is even, so we can divide by \(2\):
\[
10032 \div 2 = 5016
\]
2. **Continue factoring \(5016\)**:
\[
5016 \div 2 = 2508
\]
3. **Continue factoring \(2508\)**:
\[
2508 \div 2 = 1254
\]
4. **Continue factoring \(1254\)**:
\[
1254 \div 2 = 627
\]
5. **Now, since \(627\) is not even, check for divisibility by \(3\)** (sum of digits \(6 + 2 + 7 = 15\) which is divisible by \(3\)):
\[
627 \div 3 = 209
\]
6. **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:
\[
\boxed{(2508)^2}
\]