The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is

Options

A. 1356
B. 1365
C. 1256
D. 1465

Explanation

To find the sum of all two-digit positive numbers that yield a remainder of 2 or 5 when divided by 7, we can break the problem into two parts: one for the remainder of 2 and another for the remainder of 5.

### Step 1: Finding numbers with remainder 2
A number \( n \) that gives a remainder of 2 when divided by 7 can be expressed as:
\[ n = 7k + 2 \]
where \( k \) is a non-negative integer.

#### Finding the range of \( k \):
Since we are looking for two-digit numbers, we need:
\[ 10 \leq 7k + 2 \leq 99 \]

- For the lower bound:
\[ 7k + 2 \geq 10 \]
\[ 7k \geq 8 \]
\[ k \geq \frac{8}{7} \]
Thus, the smallest integer \( k \) is 2.

- For the upper bound:
\[ 7k + 2 \leq 99 \]
\[ 7k \leq 97 \]
\[ k \leq \frac{97}{7} \]
Thus, the largest integer \( k \) is 13.

#### Finding the two-digit numbers:
Now, we can calculate the two-digit numbers for \( k = 2 \) to \( k = 13 \):
- For \( k = 2: n = 7(2) + 2 = 16 \)
- For \( k = 3: n = 7(3) + 2 = 23 \)
- For \( k = 4: n = 7(4) + 2 = 30 \)
- For \( k = 5: n = 7(5) + 2 = 37 \)
- For \( k = 6: n = 7(6) + 2 = 44 \)
- For \( k = 7: n = 7(7) + 2 = 51 \)
- For \( k = 8: n = 7(8) + 2 = 58 \)
- For \( k = 9: n = 7(9) + 2 = 65 \)
- For \( k = 10: n = 7(10) + 2 = 72 \)
- For \( k = 11: n = 7(11) + 2 = 79 \)
- For \( k = 12: n = 7(12) + 2 = 86 \)
- For \( k = 13: n = 7(13) + 2 = 93 \)

The numbers are: 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93.

### Step 2: Sum of numbers with remainder 2
This sequence is an arithmetic progression (AP) where:
- First term \( a = 16 \)
- Last term \( l = 93 \)
- Number of terms \( n = 12 \)

The sum \( S_1 \) of an AP can be calculated using the formula:
\[ S = \frac{n}{2} \times (a + l) \]
Substituting the values:
\[ S_1 = \frac{12}{2} \times (16 + 93) = 6 \times 109 = 654 \]

### Step 3: Finding numbers with remainder 5
Similarly, a number \( n \) that gives a remainder of 5 when divided by 7 can be expressed as:
\[ n = 7k + 5 \]

#### Finding the range of \( k \):
We need:
\[ 10 \leq 7k + 5 \leq 99 \]

- For the lower bound:
\[ 7k + 5 \geq 10 \]
\[ 7k \geq 5 \]
\[ k \geq \frac{5}{7} \]
Thus, the smallest integer \( k \) is 1.

- For the upper bound:
\[ 7k + 5 \leq 99 \]
\[ 7k \leq 94 \]
\[ k \leq \frac{94}{7} \]
Thus, the largest integer \( k \) is 13.

#### Finding the two-digit numbers:
Now, we can calculate the two-digit numbers for \( k = 1 \) to \( k = 13 \):
- For \( k = 1: n = 7(1) + 5 = 12 \)
- For \( k = 2: n = 7(2) + 5 = 19 \)
- For \( k = 3: n = 7(3) + 5 = 26 \)
- For \( k = 4: n = 7(4) + 5 = 33 \)
- For \( k = 5: n = 7(5) + 5 = 40 \)
- For \( k = 6: n = 7(6) + 5 = 47 \)
- For \( k = 7: n = 7(7) + 5 = 54 \)
- For \( k = 8: n = 7(8) + 5 = 61 \)
- For \( k = 9: n = 7(9) + 5 = 68 \)
- For \( k = 10: n = 7(10) + 5 = 75 \)
- For \( k = 11: n = 7(11) + 5 = 82 \)
- For \( k = 12: n = 7(12) + 5 = 89 \)
- For \( k = 13: n = 7(13) + 5 = 96 \)

The numbers are: 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96.

### Step 4: Sum of numbers with remainder 5
This sequence is also an arithmetic progression where:
- First term \( a = 12 \)
- Last term \( l = 96 \)
- Number of terms \( n = 13 \)

The sum \( S_2 \) can be calculated using the formula:
\[ S_2 = \frac{n}{2} \times (a + l) \]
Substituting the values:
\[ S_2 = \frac{13}{2} \times (12 + 96) = \frac{13}{2} \times 108 = 13 \times 54 = 702 \]

### Step 5: Total sum
Finally, the total sum of all two-digit numbers that yield a remainder of 2 or 5 when divided by 7 is:
\[ S = S_1 + S_2 = 654 + 702 = 1356 \]

### Final Answer:
The sum of all two-digit positive numbers which when divided by 7 yield 2 or 5 as remainder is **1356**.