All possible two-factor products are formed from the numbers 1, 2,…..,100. The numbers of factors out of the total obtained which are multiple of 3, is

Options

A. 2211
B. 4950
C. 2739
D. none of these

Answer

To solve the problem of finding the number of two-factor products formed from the numbers 1 to 100 that are multiples of 3, we can follow these steps:

### Step 1: Calculate the total number of two-factor products
The total number of two-factor products can be calculated using combinations. We need to choose 2 numbers from the set {1, 2, ..., 100}. The number of ways to choose 2 numbers from 100 is given by:

\[ \text{Total two-factor products} = \binom{100}{2} = \frac{100 \times 99}{2} = 4950 \]

### Step 2: Identify the multiples of 3 from 1 to 100
Next, we need to find how many numbers between 1 and 100 are multiples of 3. The multiples of 3 in this range are 3, 6, 9, ..., 99.

To find the count of these multiples, we can use the formula for the nth term of an arithmetic sequence:

\[ \text{Last term} = 3n \leq 100 \implies n \leq \frac{100}{3} \implies n = 33 \]

Thus, there are 33 multiples of 3.

### Step 3: Calculate the non-multiples of 3
Now, we can find the number of non-multiples of 3 in the range from 1 to 100:

\[ \text{Non-multiples of 3} = 100 - 33 = 67 \]

### Step 4: Calculate the two-factor products of non-multiples of 3
Next, we need to calculate the number of two-factor products that can be formed using the non-multiples of 3. This is given by:

\[ \text{Two-factor products of non-multiples of 3} = \binom{67}{2} = \frac{67 \times 66}{2} = 2211 \]

### Step 5: Calculate the two-factor products that are multiples of 3
Finally, to find the number of two-factor products that are multiples of 3, we subtract the number of products formed by non-multiples of 3 from the total number of products:

\[ \text{Two-factor products that are multiples of 3} = \binom{100}{2} - \binom{67}{2} = 4950 - 2211 = 2739 \]

### Conclusion
Thus, the number of two-factor products formed from the numbers 1 to 100 that are multiples of 3 is **2739**.

---