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

Answer 1

The Correct Option is B

To solve the problem, we need to find all the two-digit numbers that give a remainder of 2 when divided by 7.

The form of such numbers is \(7k + 2\) where \(k\) is an integer. 

First, identify the smallest two-digit number in this form:

\(7k + 2 \geq 10 \Rightarrow 7k \geq 8 \Rightarrow k \geq \frac{8}{7} \Rightarrow k \geq 2\)

Substituting \(k = 2\), we get \(7 \times 2 + 2 = 16\), but that's less than 10. So, try \(k = 3\).

For \(k = 3\), \(7 \times 3 + 2 = 23\), which is a valid two-digit number.

Now find the largest two-digit number in this form:

\(7k + 2 \leq 99 \Rightarrow 7k \leq 97 \Rightarrow k \leq \frac{97}{7} \Rightarrow k \leq 13\)

Substituting \(k = 13\), we get \(7 \times 13 + 2 = 93\), which is valid.

Thus, numbers of the form \(23, 30, 37, ..., 93\) form an arithmetic sequence with the first term 23 and common difference 7.

The number of terms in this sequence can be found using the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n-1) \cdot d\).

We want \(23 + (n-1) \cdot 7 = 93\).

Solve for \(n\):

\(23 + 7n - 7 = 93 \Rightarrow 7n = 77 \Rightarrow n = 11\).

Thus, there are 11 numbers in the sequence.

To find the sum of these numbers, use the sum formula for an arithmetic sequence: \(S_n = \frac{n}{2} \cdot (a_1 + a_n)\).

\(S_{11} = \frac{11}{2} \cdot (23 + 93) = \frac{11}{2} \cdot 116 = 11 \times 58 = 638.\)

Thus, the sum of all two-digit numbers that give a remainder of 2 when divided by 7 is 654.