Question

The sum of all two digit numbers that give a remainder 2 when they are divided by 7 is ______

• A. 552
• B. 654
• C. 658
• D. 684

Hint:

The numbers are of the form 7k+2 between 10 and 99, forming an AP from 16 to 93; use the AP sum formula.

Answer 1

The Correct Option is B

Step 1: List the terms and split the sum into two parts.
The numbers are 16, 23, 30, ..., 93, that is $7k+2$ for $k = 2$ to $13$, twelve values of $k$ in total.

Step 2: Add up the '7k' parts and the '+2' parts separately.
Sum $= 7(2+3+4+\cdots+13) + 2 \times 12$, since there are 12 terms each contributing a '+2'.

Step 3: Find the sum of k from 2 to 13.
Sum of integers from 2 to 13 $= \frac{13 \times 14}{2} - 1 = 91 - 1 = 90$.

Step 4: Combine both parts.
Sum $= 7 \times 90 + 24 = 630 + 24 = 654$.

Final Answer:
The total is 654, the same value we get by treating the numbers as an AP. \[ \boxed{654} \]