If a number is divided by 119, the remainder is 19, what will be the remainder if the same number is divided by 17 ?

Options

A. 5
B. 0
C. 2
D. 9

Answer

To solve the problem step by step, we can follow these instructions:

1. **Define the Number**: Let the number be \( M \).

2. **Use the Given Information**: According to the problem, when \( M \) is divided by 119, the remainder is 19. This can be expressed mathematically as:
\[ M = 119k + 19 \]
where \( k \) is some integer.

3. **Find the Remainder When Divided by 17**: We need to find the remainder when \( M \) is divided by 17. We will substitute the expression for \( M \) into the division by 17:
\[ M = 119k + 19 \]

4. **Calculate \( 119 \mod 17 \)**: First, we need to find the remainder when 119 is divided by 17. We perform the division:
\[ 119 \div 17 = 7 \quad \text{(since } 7 \times 17 = 119\text{)} \]
Thus, the remainder is:
\[ 119 \mod 17 = 0 \]

5. **Calculate \( 19 \mod 17 \)**: Next, we find the remainder when 19 is divided by 17:
\[ 19 \div 17 = 1 \quad \text{(since } 1 \times 17 = 17\text{)} \]
Thus, the remainder is:
\[ 19 \mod 17 = 2 \]

6. **Combine the Results**: Since \( 119k \mod 17 = 0 \), we have:
\[ M \mod 17 = (119k + 19) \mod 17 = (0 + 19) \mod 17 = 19 \mod 17 = 2 \]

7. **Conclusion**: Therefore, the remainder when the number \( M \) is divided by 17 is:
\[ \text{Remainder} = 2 \]