All possible two factors products are formed from the numbers 1, 2, 3, 4, ....., 200. The number of factors out of total obtained which are multiples of 5 is

Options

A. 5040
B. 7180
C. 8150
D. None of these

Answer

To solve the problem of finding the number of two-factor products formed from the numbers 1 to 200 that are multiples of 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range of Numbers**:
We are given the numbers from 1 to 200.

2. **Determine the Total Number of Two-Factor Products**:
The total number of ways to choose 2 factors from 200 numbers is given by the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).
Here, \( n = 200 \) and \( r = 2 \):
\[ C(200, 2) = \frac{200 \times 199}{2} = 19900 \]

3. **Identify Multiples of 5**:
We need to find how many of these products are multiples of 5. First, we find how many multiples of 5 are there from 1 to 200.
The multiples of 5 in this range are: 5, 10, 15, ..., 200.
This forms an arithmetic progression (AP) where:
- First term \( a = 5 \)
- Common difference \( d = 5 \)
- Last term \( l = 200 \)

To find the number of terms \( n \) in this AP, we use the formula for the last term:
\[ l = a + (n-1)d \implies 200 = 5 + (n-1) \cdot 5 \]
Solving this gives:
\[ 200 - 5 = (n-1) \cdot 5 \implies 195 = (n-1) \cdot 5 \implies n-1 = 39 \implies n = 40 \]
Thus, there are 40 multiples of 5 between 1 and 200.

4. **Determine Non-Multiples of 5**:
The total numbers from 1 to 200 is 200. The non-multiples of 5 are:
\[ 200 - 40 = 160 \]

5. **Calculate the Number of Two-Factor Products that are Not Multiples of 5**:
Now we find the number of two-factor products that are not multiples of 5. This is given by:
\[ C(160, 2) = \frac{160 \times 159}{2} = 12720 \]

6. **Calculate the Number of Two-Factor Products that are Multiples of 5**:
To find the number of products that are multiples of 5, we subtract the number of products that are not multiples of 5 from the total number of products:
\[ \text{Multiples of 5} = C(200, 2) - C(160, 2) = 19900 - 12720 = 7180 \]

### Final Answer:
The number of two-factor products that are multiples of 5 is **7180**.

---