Total number of factors of a number is 24 and the sum of its 3 prime factors out of four, is 25. The product of all 4 prine factors of this number is 1365. Then such a greatest possible number can be :

Options

A. a.17745
B. b.28561
C. c.4095
D. d.can't be determined

Answer

To solve the problem step by step, we need to find a number that meets the following criteria:

1. The total number of factors is 24.
2. The sum of three prime factors out of four is 25.
3. The product of all four prime factors is 1365.

### Step 1: Factorization of 1365
First, we need to factor 1365 to find its prime factors.

- Divide 1365 by 5:
\[ 1365 \div 5 = 273 \]
- Now, factor 273:
- Divide 273 by 3:
\[ 273 \div 3 = 91 \]
- Now, factor 91:
- Divide 91 by 7:
\[ 91 \div 7 = 13 \]

So, the prime factorization of 1365 is:
\[ 1365 = 5 \times 3 \times 7 \times 13 \]

### Step 2: Verify the number of factors
To find the total number of factors, we can use the formula for the number of factors based on the prime factorization:
\[ (n_1 + 1)(n_2 + 1)(n_3 + 1)(n_4 + 1) \]
where \( n_i \) are the powers of the prime factors.

From our factorization:
\[ 1365 = 5^1 \times 3^1 \times 7^1 \times 13^1 \]
The powers are all 1, so:
\[ (1+1)(1+1)(1+1)(1+1) = 2 \times 2 \times 2 \times 2 = 16 \]
This does not satisfy the condition of having 24 factors.

### Step 3: Adjust the prime factorization
Since we need a total of 24 factors, we can adjust the powers of the prime factors. We can try increasing the power of one of the prime factors.

Let’s try:
\[ 5^3 \times 3^1 \times 7^1 \times 13^1 \]
Calculating the number of factors:
\[ (3+1)(1+1)(1+1)(1+1) = 4 \times 2 \times 2 \times 2 = 32 \]
This is too high.

Next, let’s try:
\[ 5^2 \times 3^1 \times 7^1 \times 13^1 \]
Calculating the number of factors:
\[ (2+1)(1+1)(1+1)(1+1) = 3 \times 2 \times 2 \times 2 = 24 \]
This satisfies the condition of having 24 factors.

### Step 4: Check the sum of three prime factors
Now we have the prime factors as \( 5, 3, 7, 13 \). We need to check if the sum of any three of these equals 25.

Calculating the sums:
- \( 5 + 3 + 7 = 15 \)
- \( 5 + 3 + 13 = 21 \)
- \( 5 + 7 + 13 = 25 \) (This works!)
- \( 3 + 7 + 13 = 23 \)

So, the sum of \( 5, 7, \) and \( 13 \) is indeed 25.

### Step 5: Calculate the greatest possible number
Now we can calculate the greatest possible number using the prime factorization:
\[ N = 5^2 \times 3^1 \times 7^1 \times 13^1 \]
Calculating:
\[ N = 25 \times 3 \times 7 \times 13 \]
Calculating step by step:
- \( 25 \times 3 = 75 \)
- \( 75 \times 7 = 525 \)
- \( 525 \times 13 = 6825 \)

Thus, the greatest possible number is:
\[ \boxed{6825} \]