A number x when divided by 289 leaves 18 as the remainder. The same number when divided by 17 leaves y as a remainder. The value of y is

Options

A. a) 5
B. b) 2
C. c) 3
D. d) 1

Answer

To solve the problem, we need to find the value of \( y \) when the number \( x \) is divided by 17, given that \( x \) leaves a remainder of 18 when divided by 289.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We know that when \( x \) is divided by 289, it leaves a remainder of 18. This can be expressed mathematically as:
\[ x = 289k + 18 \]
where \( k \) is some integer.

2. **Finding \( x \) Modulo 17:**
We need to find the remainder when \( x \) is divided by 17. To do this, we can substitute the expression for \( x \) into the modulo operation:
\[ x \mod 17 = (289k + 18) \mod 17 \]

3. **Calculating \( 289 \mod 17 \):**
First, we need to find \( 289 \mod 17 \):
\[ 289 \div 17 = 17 \quad \text{(since } 17 \times 17 = 289\text{)} \]
Therefore, the remainder is:
\[ 289 \mod 17 = 0 \]

4. **Substituting Back:**
Now we can substitute this back into our equation:
\[ x \mod 17 = (0 \cdot k + 18) \mod 17 = 18 \mod 17 \]

5. **Calculating \( 18 \mod 17 \):**
Now, we find \( 18 \mod 17 \):
\[ 18 \div 17 = 1 \quad \text{(since } 17 \times 1 = 17\text{)} \]
The remainder is:
\[ 18 - 17 = 1 \]

6. **Conclusion:**
Therefore, the value of \( y \) is:
\[ y = 1 \]

### Final Answer:
The value of \( y \) is \( 1 \).