Two numbers, when divided by 17, leave remainders 13 and 11 respectively. If the sum of those two numbers is divided by 17, the remainder will be

Options

A. 13
B. 11
C. 7
D. 4

Answer

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

### Step 1: Understand the problem
We have two numbers that leave specific remainders when divided by 17. The first number leaves a remainder of 13, and the second number leaves a remainder of 11.

### Step 2: Represent the numbers
Let the first number be \( x \) and the second number be \( y \). According to the problem:
- When \( x \) is divided by 17, it leaves a remainder of 13.
- When \( y \) is divided by 17, it leaves a remainder of 11.

This can be expressed mathematically as:
- \( x = 17k + 13 \) (for some integer \( k \))
- \( y = 17m + 11 \) (for some integer \( m \))

### Step 3: Find the sum of the two numbers
Now, we need to find the sum of these two numbers:
\[ x + y = (17k + 13) + (17m + 11) \]
This simplifies to:
\[ x + y = 17k + 17m + 24 \]
\[ x + y = 17(k + m) + 24 \]

### Step 4: Find the remainder when the sum is divided by 17
To find the remainder when \( x + y \) is divided by 17, we can focus on the constant term (24) since \( 17(k + m) \) will be divisible by 17 and will leave a remainder of 0.

Now we need to find the remainder of 24 when divided by 17:
\[ 24 \div 17 = 1 \quad \text{(which gives a quotient of 1)} \]
The remainder can be calculated as:
\[ 24 - 17 \times 1 = 24 - 17 = 7 \]

### Final Answer
Thus, when the sum of the two numbers is divided by 17, the remainder is **7**.

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