The sum of all the factors of 45000 which are exactly the multiples of 10 is :

Options

A. 152295
B. 141960
C. 600
D. None of these

Answer

To find the sum of all the factors of 45000 that are exactly multiples of 10, we can follow these steps:

### Step 1: Prime Factorization of 45000
First, we need to find the prime factorization of 45000.

45000 can be broken down as follows:
- 45000 = 450 × 100
- 450 = 45 × 10 = (5 × 9) × (2 × 5) = (5 × 3 × 3) × (2 × 5)
- 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2² × 5²

So, combining these, we have:
- 45000 = 5² × (3²) × (2 × 5) × (2² × 5²)
- This simplifies to: 45000 = 2³ × 3² × 5⁴

### Step 2: Identify Factors that are Multiples of 10
Next, we need to find the factors of 45000 that are multiples of 10. A number is a multiple of 10 if it includes at least one factor of 2 and one factor of 5.

From the prime factorization:
- The factors of 10 can be expressed as: 10 = 2¹ × 5¹

### Step 3: Determine the Range of Exponents for Factors
To find the factors of 45000 that are multiples of 10, we can express them in the form:
- \(2^a × 3^b × 5^c\)
where:
- \(a \geq 1\) (since we need at least one factor of 2)
- \(b\) can be 0, 1, or 2 (since the maximum exponent of 3 is 2)
- \(c \geq 1\) and can go up to 4 (since the maximum exponent of 5 is 4)

### Step 4: Calculate the Sum of the Factors
Using the formula for the sum of the factors, we can calculate it as follows:

The sum of the factors of a number in the form \(p^n\) is given by:
\[ \text{Sum} = \frac{p^{n+1} - 1}{p - 1} \]

For our case:
1. For \(2^a\) where \(a = 1, 2, 3\):
\[ \text{Sum} = 2^1 + 2^2 + 2^3 = 2 + 4 + 8 = 14 \]

2. For \(3^b\) where \(b = 0, 1, 2\):
\[ \text{Sum} = 3^0 + 3^1 + 3^2 = 1 + 3 + 9 = 13 \]

3. For \(5^c\) where \(c = 1, 2, 3, 4\):
\[ \text{Sum} = 5^1 + 5^2 + 5^3 + 5^4 = 5 + 25 + 125 + 625 = 780 \]

### Step 5: Combine the Sums
Now, we multiply the sums together:
\[ \text{Total Sum} = 14 × 13 × 780 \]

Calculating this:
1. \(14 × 13 = 182\)
2. \(182 × 780 = 141960\)

### Final Answer
Thus, the sum of all the factors of 45000 that are exactly multiples of 10 is **141960**.

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