What is the sum of all the two-digit numbers which when divided by 7 gives a remainder of 3?

Options

A. 94
B. 676
C. 696
D. None of these

Answer

To find the sum of all two-digit numbers that give a remainder of 3 when divided by 7, we can follow these steps:

### Step 1: Identify the smallest two-digit number that gives a remainder of 3 when divided by 7.
The smallest two-digit number is 10. When we divide 10 by 7, we get:
\[ 10 \div 7 = 1 \quad \text{(remainder 3)} \]
So, the smallest two-digit number that gives a remainder of 3 when divided by 7 is **10**.

### Step 2: Identify the largest two-digit number that gives a remainder of 3 when divided by 7.
The largest two-digit number is 99. When we divide 99 by 7, we get:
\[ 99 \div 7 = 14 \quad \text{(remainder 1)} \]
To find the largest two-digit number that gives a remainder of 3, we can subtract 4 from 99:
\[ 99 - 4 = 95 \]
Now, we check:
\[ 95 \div 7 = 13 \quad \text{(remainder 4)} \]
This is incorrect. We try the next number down, which is 94:
\[ 94 \div 7 = 13 \quad \text{(remainder 3)} \]
So, the largest two-digit number that gives a remainder of 3 when divided by 7 is **94**.

### Step 3: List the two-digit numbers that give a remainder of 3 when divided by 7.
The two-digit numbers that give a remainder of 3 when divided by 7 form an arithmetic sequence:
- First term (a) = 10
- Common difference (d) = 7
- Last term (l) = 94

The sequence is: 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94.

### Step 4: Determine the number of terms (n) in the sequence.
To find the number of terms in the sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 94 = 10 + (n - 1) \cdot 7 \]
\[ 94 - 10 = (n - 1) \cdot 7 \]
\[ 84 = (n - 1) \cdot 7 \]
\[ n - 1 = \frac{84}{7} \]
\[ n - 1 = 12 \]
\[ n = 13 \]

### Step 5: Calculate the sum of the arithmetic sequence.
The sum \( S_n \) of the first n terms of an arithmetic sequence can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the known values:
\[ S_{13} = \frac{13}{2} \cdot (10 + 94) \]
\[ S_{13} = \frac{13}{2} \cdot 104 \]
\[ S_{13} = 13 \cdot 52 \]
\[ S_{13} = 676 \]

### Final Answer:
The sum of all the two-digit numbers which when divided by 7 gives a remainder of 3 is **676**.

---