A positive integer when divided by 425 gives a remainder 45. When the same number is divided by 17, the remainder will be.

Options

A. 10
B. 11
C. 8
D. 9

Explanation

To solve the problem step by step, we will follow the logic presented in the video transcript while ensuring clarity in each step.

### Step 1: Understand the given condition
We are given that a positive integer \( N \) when divided by 425 gives a remainder of 45. This can be expressed mathematically as:
\[ N = 425k + 45 \]
where \( k \) is some integer.

### Step 2: Rewrite the expression for \( N \)
Now, we need to find the remainder when \( N \) is divided by 17. We can substitute our expression for \( N \):
\[ N = 425k + 45 \]

### Step 3: Find \( 425 \mod 17 \)
Next, we need to calculate \( 425 \mod 17 \):
- First, divide 425 by 17:
\[ 425 \div 17 = 25 \]
- Now, multiply 25 by 17 to find the closest multiple of 17:
\[ 25 \times 17 = 425 \]
- Now, subtract this from 425 to find the remainder:
\[ 425 - 425 = 0 \]
Thus, \( 425 \equiv 0 \mod 17 \).

### Step 4: Substitute back into the expression for \( N \)
Now we can substitute this back into our expression for \( N \):
\[ N = 425k + 45 \]
Since \( 425 \equiv 0 \mod 17 \), we have:
\[ N \equiv 0 \cdot k + 45 \mod 17 \]
This simplifies to:
\[ N \equiv 45 \mod 17 \]

### Step 5: Find \( 45 \mod 17 \)
Now we need to calculate \( 45 \mod 17 \):
- Divide 45 by 17:
\[ 45 \div 17 = 2 \] (since \( 17 \times 2 = 34 \))
- Now, subtract this from 45:
\[ 45 - 34 = 11 \]
Thus, \( 45 \equiv 11 \mod 17 \).

### Conclusion
Therefore, when the number \( N \) is divided by 17, the remainder is:
\[ \text{Remainder} = 11 \]

### Final Answer
The remainder when the number is divided by 17 is **11**.

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