How many numbers between -55 and 55 including both the extreme values are divisible by 5 ?

Options

A. 100
B. 111
C. 101
D. None of these

Answer

To find how many numbers between -55 and 55 (including both extremes) are divisible by 5, we can follow these steps:

### Step 1: Identify the range
We need to consider the numbers from -55 to 55, inclusive.

### Step 2: Identify the first and last numbers divisible by 5
- The first number in this range that is divisible by 5 is -55.
- The last number in this range that is divisible by 5 is 55.

### Step 3: List the numbers divisible by 5
The numbers divisible by 5 in this range can be listed as follows:
- Starting from -55: -55, -50, -45, -40, -35, -30, -25, -20, -15, -10, -5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55.

### Step 4: Identify the common difference
The numbers we listed form an arithmetic progression (AP) where:
- The first term (a) is -55.
- The common difference (d) is 5.

### Step 5: Use the formula for the nth term of an AP
The nth term of an AP can be calculated using the formula:
\[ a_n = a + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 6: Set up the equation for the last term
We know that the last term \( a_n \) is 55. So we set up the equation:
\[ 55 = -55 + (n - 1) \cdot 5 \]

### Step 7: Solve for n
1. Rearranging the equation:
\[ 55 + 55 = (n - 1) \cdot 5 \]
\[ 110 = (n - 1) \cdot 5 \]
2. Dividing both sides by 5:
\[ n - 1 = \frac{110}{5} \]
\[ n - 1 = 22 \]
3. Adding 1 to both sides:
\[ n = 22 + 1 \]
\[ n = 23 \]

### Conclusion
Thus, there are **23 numbers** between -55 and 55 (including both extremes) that are divisible by 5.

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