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 can follow these steps:

### Step 1: Define the two numbers
Let the two numbers be \( n_1 \) and \( n_2 \). According to the problem, when \( n_1 \) is divided by 17, it leaves a remainder of 13. This can be expressed as:
\[ n_1 = 17q + 13 \]
where \( q \) is some integer quotient.

### Step 2: Define the second number
Similarly, when \( n_2 \) is divided by 17, it leaves a remainder of 11. This can be expressed as:
\[ n_2 = 17p + 11 \]
where \( p \) is another integer quotient.

### Step 3: Find the sum of the two numbers
Now, we will find the sum of \( n_1 \) and \( n_2 \):
\[ n_1 + n_2 = (17q + 13) + (17p + 11) \]
Combining like terms, we get:
\[ n_1 + n_2 = 17q + 17p + 13 + 11 \]
\[ n_1 + n_2 = 17(q + p) + 24 \]

### Step 4: Determine the remainder when the sum is divided by 17
Now, we need to find the remainder when \( n_1 + n_2 \) is divided by 17. We can express this as:
\[ n_1 + n_2 = 17(q + p) + 24 \]
When we divide \( n_1 + n_2 \) by 17, the term \( 17(q + p) \) is completely divisible by 17, so we only need to consider the remainder of 24 when divided by 17.

### Step 5: Calculate the remainder
Now, we calculate \( 24 \div 17 \):
\[ 24 = 17 \times 1 + 7 \]
Thus, the remainder when 24 is divided by 17 is 7.

### Final Answer
Therefore, the remainder when the sum of the two numbers is divided by 17 is:
\[ \boxed{7} \]

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