If a number is divided by 102, the re- mainder is obtained 91. If the same num- ber is divided by 1 7, then remainder will be?

Options

A. 6
B. 11
C. 0
D. 2

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the given information
We know that when a certain number \( x \) is divided by 102, the remainder is 91. This can be expressed mathematically as:
\[ x \equiv 91 \mod 102 \]

### Step 2: Express the number in terms of 102
From the information above, we can express \( x \) in terms of 102:
\[ x = 102k + 91 \]
where \( k \) is some integer (the quotient when \( x \) is divided by 102).

### Step 3: Substitute a value for \( k \)
To find a specific value for \( x \), we can choose \( k = 1 \) (the smallest positive integer):
\[ x = 102(1) + 91 = 102 + 91 = 193 \]

### Step 4: Find the remainder when \( x \) is divided by 17
Now we need to find the remainder when \( x = 193 \) is divided by 17. We can perform the division:
\[ 193 \div 17 \]
Calculating this gives:
\[ 193 = 17 \times 11 + r \]
where \( r \) is the remainder. We can calculate \( 17 \times 11 = 187 \).

### Step 5: Calculate the remainder
Now, we subtract 187 from 193 to find the remainder:
\[ r = 193 - 187 = 6 \]

### Conclusion
Thus, the remainder when the number \( x \) is divided by 17 is:
\[ \text{Remainder} = 6 \]