What is the square of the product of all the factors of 53900?

2 ^ 110 * 5 ^ 110 * 7 ^ 110 * 11 ^ 55

Ο 2 ^ 106 * 5 ^ 106 * 7 ^ 106 * 11 ^ 53

2 ^ 108 * 5 ^ 108 * 7 ^ 108 * 11 ^ 54

2 ^ 112 * 5 ^ 112 * 7 ^ 112 * 11 ^ 56

Let's begin by finding the factors of \( 53900 \).

### Prime Factorization of \( 53900 \):
We first break down \( 53900 \) into prime factors.

\[
53900 = 2^2 \times 5^2 \times 7^1 \times 11^1
\]

Now, the product of all the factors of a number is given by the square of the number's divisors, since for any divisor \( d \), there is a corresponding divisor \( \frac{n}{d} \).

1. To find the product of all divisors, we first compute the total number of divisors:
   For a number \( n = 2^a \times 5^b \times 7^c \times 11^d \), the number of divisors is:

   \[
   \text{Number of divisors} = (a+1)(b+1)(c+1)(d+1)
   \]
   For \( 53900 = 2^2 \times 5^2 \times 7^1 \times 11^1 \), we have:

   \[
   \text{Number of divisors} = (2+1)(2+1)(1+1)(1+1) = 3 \times 3 \times 2 \times 2 = 36
   \]

2. Next, the product of all divisors is \( 53900^{36/2} = 53900^{18} \), because the product is the square root of \( 53900^{36} \).

### Prime factorizing \( 53900^{18} \):

Since \( 53900 = 2^2 \times 5^2 \times 7 \times 11 \), we raise each prime factor to the power of 18:

\[
53900^{18} = (2^2 \times 5^2 \times 7 \times 11)^{18} = 2^{36} \times 5^{36} \times 7^{18} \times 11^{18}
\]

### Squaring the product of all divisors:

The square of the product of all divisors is the square of \( 53900^{18} \), which is:

\[
(53900^{18})^2 = 53900^{36} = 2^{72} \times 5^{72} \times 7^{36} \times 11^{36}
\]

Thus, the square of the product of all factors of 53900 is:

\[
2^{72} \times 5^{72} \times 7^{36} \times 11^{36}
\]

None of the provided options exactly match this result, which suggests a deeper review or a slightly different interpretation is required. However, based on typical formulations of the divisor function and product of divisors, this seems to be the most consistent factorization.

Would you like more details on this or another related explanation?

### 5 Related Questions:
1. How do you determine the prime factorization of a large number?
2. What is the formula for the number of divisors of a number?
3. How is the product of all divisors of a number derived mathematically?
4. What are other common divisor-related problems in number theory?
5. How can prime factorizations help solve exponential equations?

### Tip:
Always break down composite numbers into their prime factors to simplify exponentiation problems, as this reveals patterns more easily.

What is the square of the product of all the factors of 53900?
2 ^ 110 * 5 ^ 110 * 7 ^ 110 * 11 ^ 55
Ο 2 ^ 106 * 5 ^ 106 * 7 ^ 106 * 11 ^ 53
2 ^ 108 * 5 ^ 108 * 7 ^ 108 * 11 ^ 54
2 ^ 112 * 5 ^ 112 * 7 ^ 112 * 11 ^ 56

Explore the solution for calculating the square of the product of all factors of 53900 using prime factorization. This problem covers key mathematical concepts such as prime factorization, divisor theorems, and exponentiation, making it suitable for Grades 10-12.

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