If a and b are any two distinct numbers belonging to the set {1,2,......100}, then the number of pairs (a, b) such that product of a and b is divisible by 3 is 5478 2. 5278 3. 2739 3. 2639 5. 2837

Options

A. `5478`
B. `5278`
C. `2739`
D. `2639`

Explanation

To solve the problem of finding the number of pairs (a, b) such that the product of a and b is divisible by 3, we can follow these steps:

### Step 1: Identify the total number of elements
The set consists of numbers from 1 to 100. Therefore, the total number of distinct numbers in the set is:
\[ n = 100 \]

### Step 2: Count the numbers divisible by 3
The numbers divisible by 3 in the range from 1 to 100 are:
\[ 3, 6, 9, \ldots, 99 \]
This forms an arithmetic sequence where:
- First term \( a = 3 \)
- Common difference \( d = 3 \)
- Last term \( l = 99 \)

To find the number of terms (n), we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n-1)d \]
Solving for \( n \):
\[ 99 = 3 + (n-1) \cdot 3 \]
\[ 99 - 3 = (n-1) \cdot 3 \]
\[ 96 = (n-1) \cdot 3 \]
\[ n - 1 = 32 \]
\[ n = 33 \]

So, there are 33 numbers divisible by 3.

### Step 3: Count the numbers not divisible by 3
The total numbers from 1 to 100 is 100. The numbers not divisible by 3 can be calculated as:
\[ 100 - 33 = 67 \]

### Step 4: Calculate the pairs (a, b) such that the product ab is divisible by 3
The product \( ab \) is divisible by 3 if at least one of the numbers \( a \) or \( b \) is divisible by 3. We can use complementary counting to find the pairs where \( ab \) is not divisible by 3.

#### Step 4.1: Count pairs where both a and b are not divisible by 3
The number of ways to choose 2 distinct numbers from the 67 numbers that are not divisible by 3 is given by:
\[ \binom{67}{2} = \frac{67 \cdot 66}{2} = 2211 \]

#### Step 4.2: Calculate total pairs (a, b)
The total number of ways to choose 2 distinct numbers from 100 is:
\[ \binom{100}{2} = \frac{100 \cdot 99}{2} = 4950 \]

#### Step 4.3: Calculate pairs where the product is divisible by 3
The pairs where the product \( ab \) is divisible by 3 can be found by subtracting the pairs where \( ab \) is not divisible by 3 from the total pairs:
\[ \text{Pairs where } ab \text{ is divisible by 3} = \binom{100}{2} - \binom{67}{2} \]
\[ = 4950 - 2211 = 2739 \]

### Conclusion
The number of pairs (a, b) such that the product \( ab \) is divisible by 3 is:
\[ \boxed{2739} \]