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, let's break it down into steps.

### Step 1: Identify all factors of 72
We first find all factors of 72. The prime factorization of 72 is:

\[
72 = 2^3 \times 3^2
\]

Using this, the number of factors can be calculated by using the formula for total number of factors:

\[
\text{Total factors} = (3+1)(2+1) = 4 \times 3 = 12
\]

The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

### Step 2: Calculate the sum of all factors of 72
The formula to find the sum of factors of a number is:

\[
\text{Sum of factors} = \left( \frac{p_1^{e_1+1} - 1}{p_1 - 1} \right) \times \left( \frac{p_2^{e_2+1} - 1}{p_2 - 1} \right)
\]

For 72, we apply this to the prime factorization \(72 = 2^3 \times 3^2\):

\[
\text{Sum of factors} = \left( \frac{2^{3+1} - 1}{2 - 1} \right) \times \left( \frac{3^{2+1} - 1}{3 - 1} \right)
= \left( \frac{16 - 1}{1} \right) \times \left( \frac{27 - 1}{2} \right)
= 15 \times 13 = 195
\]

Thus, \( A = 195 \).

### Step 3: Find the product of all factors of A
The number 195 can be factored as:

\[
195 = 3 \times 5 \times 13
\]

Now, we need to find the product of all factors of 195. First, calculate how many factors 195 has. Using the exponents of its prime factorization, we get:

\[
\text{Total factors of 195} = (1+1)(1+1)(1+1) = 2 \times 2 \times 2 = 8
\]

The factors of 195 are: 1, 3, 5, 13, 15, 39, 65, 195.

### Step 4: Calculate the product of all factors of A (i.e., 195)
The product of all factors of any number \( n \) is given by the formula:

\[
\text{Product of all factors of } n = n^{\frac{t(n)}{2}}
\]

Where \( t(n) \) is the total number of factors of \( n \). For \( n = 195 \) and \( t(n) = 8 \):

\[
\text{Product of all factors of } 195 = 195^{\frac{8}{2}} = 195^4
\]

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

### Final Answer:
The correct option is **(2) \( (195)^4 \)**.

Would you like further details or explanations on any part of the solution?

Here are five related questions to expand your understanding:
1. What is the general formula for the sum of divisors of a number?
2. How do you determine the number of divisors of a given number?
3. Can you explain why the product of the divisors formula is \( n^{t(n)/2} \)?
4. How would the solution change if 72 were a different number, such as 84?
5. What is the relationship between prime factorization and finding divisors?

**Tip:** When factoring large numbers, always begin by identifying small primes like 2, 3, 5 for quicker calculations!

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

Explore how to calculate the product of all factors of A, where A is the sum of factors of 72. This problem involves number theory concepts like prime factorization, the sum of divisors formula, and the product of divisors theorem. Suitable for Grades 10-12 students.

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