The sum of all two digit numbers which when divided by 4 leave 1 as remainder is

Options

A. 1210
B. 1120
C. 1021
D. 1102

Answer

To find the sum of all two-digit numbers that leave a remainder of 1 when divided by 4, we can follow these steps:

### Step 1: Identify the first two-digit number
The smallest two-digit number is 10. To find the smallest two-digit number that leaves a remainder of 1 when divided by 4, we can check the numbers:
- 10 ÷ 4 = 2 remainder 2
- 11 ÷ 4 = 2 remainder 3
- 12 ÷ 4 = 3 remainder 0
- 13 ÷ 4 = 3 remainder 1

Thus, the first two-digit number that leaves a remainder of 1 when divided by 4 is **13**.

### Step 2: Identify the last two-digit number
The largest two-digit number is 99. To find the largest two-digit number that leaves a remainder of 1 when divided by 4, we can check the numbers:
- 99 ÷ 4 = 24 remainder 3
- 98 ÷ 4 = 24 remainder 2
- 97 ÷ 4 = 24 remainder 1

Thus, the last two-digit number that leaves a remainder of 1 when divided by 4 is **97**.

### Step 3: List the sequence of numbers
The two-digit numbers that leave a remainder of 1 when divided by 4 form an arithmetic progression (AP):
- First term (a) = 13
- Common difference (d) = 4 (since the next number is 13 + 4 = 17, and so on)
- Last term (l) = 97

The sequence is: 13, 17, 21, ..., 97.

### Step 4: Find the number of terms (n)
To find the number of terms in the AP, we use the formula for the nth term of an AP:
\[ a_n = a + (n - 1) \cdot d \]

Setting \( a_n = 97 \):
\[ 97 = 13 + (n - 1) \cdot 4 \]

Now, solve for n:
1. Subtract 13 from both sides:
\[ 97 - 13 = (n - 1) \cdot 4 \]
\[ 84 = (n - 1) \cdot 4 \]
2. Divide by 4:
\[ n - 1 = 21 \]
3. Add 1:
\[ n = 22 \]

### Step 5: Calculate the sum of the AP
The sum of the first n terms of an AP is given by:
\[ S_n = \frac{n}{2} \cdot (a + l) \]

Substituting the values:
- \( n = 22 \)
- \( a = 13 \)
- \( l = 97 \)

\[ S_{22} = \frac{22}{2} \cdot (13 + 97) \]
\[ S_{22} = 11 \cdot 110 \]
\[ S_{22} = 1210 \]

### Final Answer
The sum of all two-digit numbers which when divided by 4 leave a remainder of 1 is **1210**.

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