Direction : s_(1) = 1; s_(2) = 2,3; s_(3) = 4, 5, 6; s_(4) = 7, 8, 9,10; s_(5) = 11, 12, 13, 14, 15....etc. where s_(1), s_(2) ,s_(3),s_(4) ,s_(5) … etc. are the first second and third terms ….. of the given sequence. The largest power of 10 that can exactly divide the product of all the elements of `s_(19)` and `s_(20)` is :

Options

A. A. 10
B. B. 9
C. C. 19
D. D. can't be determined

Answer

To solve the problem, we need to find the largest power of 10 that can exactly divide the product of all the elements of \( s_{19} \) and \( s_{20} \).

### Step 1: Identify the elements in \( s_{19} \) and \( s_{20} \)

The sequence \( s_n \) consists of \( n \) consecutive integers starting from a specific point. The first element of \( s_n \) can be determined by the formula:

\[ \text{First element of } s_n = \frac{n(n-1)}{2} + 1 \]

Using this formula, we can find the first elements of \( s_{19} \) and \( s_{20} \):

- For \( s_{19} \):
\[ \text{First element of } s_{19} = \frac{19 \times 18}{2} + 1 = 171 \]
Thus, \( s_{19} = 171, 172, 173, \ldots, 189 \) (19 elements).

- For \( s_{20} \):
\[ \text{First element of } s_{20} = \frac{20 \times 19}{2} + 1 = 191 \]
Thus, \( s_{20} = 191, 192, 193, \ldots, 210 \) (20 elements).

### Step 2: Calculate the product of all elements in \( s_{19} \) and \( s_{20} \)

The product we need to consider is:
\[ P = \prod_{k=171}^{189} k \times \prod_{k=191}^{210} k \]

### Step 3: Count the factors of 2 and 5 in \( P \)

To find the largest power of 10 that divides \( P \), we need to find the minimum of the number of factors of 2 and 5 in \( P \) since \( 10 = 2 \times 5 \).

#### Count the factors of 5

To count the number of factors of 5 in \( P \):
1. Count multiples of 5 in \( s_{19} \) (171 to 189):
- Multiples of 5: 175, 180, 185
- Count: 3

2. Count multiples of 5 in \( s_{20} \) (191 to 210):
- Multiples of 5: 195, 200, 205, 210
- Count: 4

Total factors of 5 in \( P \):
\[ 3 + 4 = 7 \]

#### Count the factors of 2

To count the number of factors of 2 in \( P \):
1. Count multiples of 2 in \( s_{19} \) (171 to 189):
- Even numbers: 172, 174, 176, 178, 180, 182, 184, 186, 188
- Count: 9

2. Count multiples of 2 in \( s_{20} \) (191 to 210):
- Even numbers: 192, 194, 196, 198, 200, 202, 204, 206, 208, 210
- Count: 10

Total factors of 2 in \( P \):
\[ 9 + 10 = 19 \]

### Step 4: Determine the largest power of 10

The largest power of 10 that divides \( P \) is given by the minimum of the counts of factors of 2 and 5:
\[ \text{Largest power of } 10 = \min(19, 7) = 7 \]

### Final Answer

The largest power of 10 that can exactly divide the product of all the elements of \( s_{19} \) and \( s_{20} \) is \( 10^7 \).

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