The remainder when the positive integer m is divided by n is r. What is the remainder when 2m is divided by 2n?

Options

A. `r`
B. `2r`
C. `2n`
D. `m - nr`

Answer

To solve the problem step by step, we will use the division algorithm and the information given in the question.

### Step 1: Understand the division algorithm
According to the division algorithm, when a positive integer \( m \) is divided by \( n \), it can be expressed as:
\[ m = nq + r \]
where:
- \( n \) is the divisor,
- \( q \) is the quotient,
- \( r \) is the remainder (with \( 0 \leq r < n \)).

### Step 2: Express \( m \) in terms of \( n \), \( q \), and \( r \)
From the division algorithm, we can write:
\[ m = nk + r \]
where \( k \) is some integer (the quotient when \( m \) is divided by \( n \)).

### Step 3: Calculate \( 2m \)
Now, we need to find \( 2m \):
\[ 2m = 2(nk + r) = 2nk + 2r \]

### Step 4: Divide \( 2m \) by \( 2n \)
Next, we will divide \( 2m \) by \( 2n \):
\[ 2m = 2nk + 2r \]
Here, \( 2n \) is the divisor.

### Step 5: Identify the quotient and remainder
When dividing \( 2m \) by \( 2n \):
- The term \( 2nk \) is completely divisible by \( 2n \), so it contributes to the quotient.
- The term \( 2r \) is what remains after dividing \( 2nk \) by \( 2n \).

Thus, the remainder when \( 2m \) is divided by \( 2n \) is:
\[ \text{Remainder} = 2r \]

### Conclusion
Therefore, the remainder when \( 2m \) is divided by \( 2n \) is \( 2r \).

### Final Answer
The answer is \( 2r \).

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