If A is the sum of the factors of 72. Then what
is the product of all the factors of A?
(1) (195)8
(2) (195)4
(3) 1950 (4) 72A

To solve the problem, we first need to find the sum of the factors of 72 (denoted as \( A \)) and then the product of all the factors of \( A \).

### Step 1: Find the prime factorization of 72
The prime factorization of 72 is:
\[
72 = 2^3 \times 3^2
\]

### Step 2: Use the formula for the sum of the factors
The formula for the sum of the factors of a number \( n \) is given by:
\[
\sigma(n) = (1 + p_1 + p_1^2 + \cdots + p_1^{k_1})(1 + p_2 + p_2^2 + \cdots + p_2^{k_2}) \cdots
\]
where \( p_1, p_2, \dots \) are the prime factors, and \( k_1, k_2, \dots \) are their respective powers in the prime factorization of \( n \).

For \( n = 72 \), we apply the formula:
\[
\sigma(72) = (1 + 2 + 2^2 + 2^3)(1 + 3 + 3^2)
\]

Simplifying:
\[
1 + 2 + 4 + 8 = 15
\]
\[
1 + 3 + 9 = 13
\]
\[
\sigma(72) = 15 \times 13 = 195
\]

Thus, \( A = 195 \).

### Step 3: Find the product of all factors of \( A \)
The product of all the factors of a number \( n \) is given by \( n^{t(n)/2} \), where \( t(n) \) is the number of divisors of \( n \).

### Step 4: Find the number of divisors of \( A = 195 \)
The prime factorization of 195 is:
\[
195 = 3 \times 5 \times 13
\]
The number of divisors \( t(n) \) is found by adding 1 to the exponents in the prime factorization and then multiplying:
\[
t(195) = (1+1)(1+1)(1+1) = 2 \times 2 \times 2 = 8
\]

### Step 5: Compute the product of all divisors
The product of all divisors of \( A = 195 \) is:
\[
195^{t(195)/2} = 195^{8/2} = 195^4
\]

Thus, the product of all the factors of \( A \) is \( 195^4 \).

### Final Answer:
The correct answer is:
\[
\boxed{(195)^4}
\]

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Do you want more details or have any questions?  
Here are 5 related questions for further practice:

1. What is the sum of the factors of 100?
2. How many divisors does 180 have?
3. What is the product of the factors of 48?
4. Find the sum of divisors of 144.
5. How do you calculate the product of the factors of a prime number?

**Tip:** The sum of divisors formula can help simplify divisor-related problems, especially when working with factorized numbers.

If A is the sum of the factors of 72, then what is the product of all the factors of A? (1) (195)^8 (2) (195)^4 (3) 1950 (4) 72A

This detailed guide helps you solve a number theory problem by finding the sum and product of divisors. We calculate the sum of divisors of 72 and the product of all divisors of 195. Suitable for Grades 9-12, this example covers prime factorization and divisor theorems.

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