The `S = {(1,3,5,7,9,…..99)(102,104,106,…,200)}` i.e., in the first part there are odd integers less than 100 and in the second part there are even integers greater than 100, but upto 200.
The highest power of 3 in the product of the element of the set is :

Options

A. 52
B. 51
C. 97
D. can't be determined

Answer

To find the highest power of 3 in the product of the elements of the set \( S = \{ (1, 3, 5, \ldots, 99), (102, 104, 106, \ldots, 200) \} \), we will break the problem into two parts: the odd integers less than 100 and the even integers greater than 100 but up to 200.

### Step 1: Count the odd integers less than 100
The odd integers less than 100 are given by the sequence:
\[ 1, 3, 5, \ldots, 99 \]
This is an arithmetic sequence where:
- First term \( a = 1 \)
- Common difference \( d = 2 \)
- Last term \( l = 99 \)

To find the number of terms \( n \) in this sequence, we can use the formula:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{99 - 1}{2} + 1 = \frac{98}{2} + 1 = 49 + 1 = 50 \]

### Step 2: Find the highest power of 3 in the odd integers
Next, we need to find how many times 3 appears as a factor in the product of these odd integers. We can do this by counting the multiples of powers of 3.

1. **Multiples of \( 3^1 = 3 \)**:
The odd multiples of 3 less than 100 are:
\[ 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99 \]
This forms an arithmetic sequence with:
- First term \( a = 3 \)
- Common difference \( d = 6 \)
- Last term \( l = 99 \)

The number of terms is:
\[ n = \frac{99 - 3}{6} + 1 = \frac{96}{6} + 1 = 16 + 1 = 17 \]

2. **Multiples of \( 3^2 = 9 \)**:
The odd multiples of 9 less than 100 are:
\[ 9, 27, 45, 63, 81, 99 \]
This forms an arithmetic sequence with:
- First term \( a = 9 \)
- Common difference \( d = 18 \)
- Last term \( l = 99 \)

The number of terms is:
\[ n = \frac{99 - 9}{18} + 1 = \frac{90}{18} + 1 = 5 + 1 = 6 \]

3. **Multiples of \( 3^3 = 27 \)**:
The odd multiples of 27 less than 100 are:
\[ 27, 81 \]
This forms an arithmetic sequence with:
- First term \( a = 27 \)
- Common difference \( d = 54 \)
- Last term \( l = 81 \)

The number of terms is:
\[ n = \frac{81 - 27}{54} + 1 = \frac{54}{54} + 1 = 1 + 1 = 2 \]

4. **Multiples of \( 3^4 = 81 \)**:
The only odd multiple of 81 less than 100 is:
\[ 81 \]
So, there is 1 term.

Now, we can sum these counts to find the total contribution of 3 from the odd integers:
- From \( 3^1 \): 17
- From \( 3^2 \): 6
- From \( 3^3 \): 2
- From \( 3^4 \): 1

Total contribution from odd integers:
\[ 17 + 6 + 2 + 1 = 26 \]

### Step 3: Count the even integers greater than 100 and up to 200
The even integers greater than 100 and up to 200 are given by the sequence:
\[ 102, 104, 106, \ldots, 200 \]
This is an arithmetic sequence where:
- First term \( a = 102 \)
- Common difference \( d = 2 \)
- Last term \( l = 200 \)

To find the number of terms \( n \):
\[ n = \frac{200 - 102}{2} + 1 = \frac{98}{2} + 1 = 49 + 1 = 50 \]

### Step 4: Find the highest power of 3 in the even integers
1. **Multiples of \( 3^1 = 3 \)**:
The even multiples of 3 between 102 and 200 are:
\[ 102, 108, 114, \ldots, 198 \]
This forms an arithmetic sequence with:
- First term \( a = 102 \)
- Common difference \( d = 6 \)
- Last term \( l = 198 \)

The number of terms is:
\[ n = \frac{198 - 102}{6} + 1 = \frac{96}{6} + 1 = 16 + 1 = 17 \]

2. **Multiples of \( 3^2 = 9 \)**:
The even multiples of 9 between 102 and 200 are:
\[ 108, 126, 144, 162, 180, 198 \]
This forms an arithmetic sequence with:
- First term \( a = 108 \)
- Common difference \( d = 18 \)
- Last term \( l = 198 \)

The number of terms is:
\[ n = \frac{198 - 108}{18} + 1 = \frac{90}{18} + 1 = 5 + 1 = 6 \]

3. **Multiples of \( 3^3 = 27 \)**:
The even multiples of 27 between 102 and 200 are:
\[ 108, 162 \]
This forms an arithmetic sequence with:
- First term \( a = 108 \)
- Common difference \( d = 54 \)
- Last term \( l = 162 \)

The number of terms is:
\[ n = \frac{162 - 108}{54} + 1 = \frac{54}{54} + 1 = 1 + 1 = 2 \]

4. **Multiples of \( 3^4 = 81 \)**:
The only even multiple of 81 between 102 and 200 is:
\[ 162 \]
So, there is 1 term.

Now, we can sum these counts to find the total contribution of 3 from the even integers:
- From \( 3^1 \): 17
- From \( 3^2 \): 6
- From \( 3^3 \): 2
- From \( 3^4 \): 1

Total contribution from even integers:
\[ 17 + 6 + 2 + 1 = 26 \]

### Step 5: Combine contributions from both parts
Finally, we sum the contributions from the odd and even integers:
\[ 26 \text{ (from odd integers)} + 26 \text{ (from even integers)} = 52 \]

### Conclusion
The highest power of 3 in the product of the elements of the set \( S \) is:
\[ \boxed{52} \]