How many distinct natural numbers are the divisors of `(30) ^(4)` ?

Options

A. A)125
B. B)123
C. C)122
D. D)124

Answer

To find the number of distinct natural numbers that are the divisors of \( (30)^4 \), we can follow these steps:

### Step 1: Prime Factorization of 30
First, we need to factor 30 into its prime factors.
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

### Step 2: Raise Each Factor to the Power of 4
Since we are interested in \( (30)^4 \), we raise each prime factor to the power of 4:
\[ (30)^4 = (2^1 \times 3^1 \times 5^1)^4 = 2^4 \times 3^4 \times 5^4 \]

### Step 3: Use the Divisor Formula
To find the total number of divisors of a number given its prime factorization \( p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \), the formula is:
\[ \text{Number of Divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
In our case, the exponents \( e_1, e_2, \) and \( e_3 \) are all 4 (from \( 2^4, 3^4, 5^4 \)).

### Step 4: Calculate the Number of Divisors
Now we can apply the formula:
\[ \text{Number of Divisors} = (4 + 1)(4 + 1)(4 + 1) = 5 \times 5 \times 5 \]
Calculating this gives:
\[ 5 \times 5 = 25 \]
\[ 25 \times 5 = 125 \]

### Conclusion
Thus, the total number of distinct natural numbers that are the divisors of \( (30)^4 \) is \( 125 \).

### Final Answer
The answer is \( 125 \).

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