Find the total numbers between 100 and 200 which are divisible by 12.

Options

A. 6
B. 16
C. 8
D. 12

Answer

To solve the problem of finding the total numbers between 100 and 200 that are divisible by 12, we can follow these steps:

### Step 1: Identify the range
We need to find numbers between 100 and 200 that are divisible by 12.

### Step 2: Find the smallest number greater than 100 that is divisible by 12
To find the smallest number greater than 100 that is divisible by 12, we can divide 100 by 12 and round up to the nearest whole number, then multiply by 12.

\[ 100 \div 12 = 8.33 \quad \text{(round up to 9)} \]
\[ 9 \times 12 = 108 \]

So, the smallest number greater than 100 that is divisible by 12 is **108**.

### Step 3: Find the largest number less than 200 that is divisible by 12
Next, we find the largest number less than 200 that is divisible by 12 by dividing 200 by 12 and rounding down to the nearest whole number, then multiplying by 12.

\[ 200 \div 12 = 16.67 \quad \text{(round down to 16)} \]
\[ 16 \times 12 = 192 \]

So, the largest number less than 200 that is divisible by 12 is **192**.

### Step 4: List the numbers divisible by 12 between 108 and 192
The numbers divisible by 12 between 108 and 192 form an arithmetic progression (AP) where:
- First term (a) = 108
- Last term (l) = 192
- Common difference (d) = 12

### Step 5: Find the number of terms (n) in the AP
The formula for the nth term of an AP is given by:
\[ l = a + (n-1)d \]
Rearranging gives:
\[ n = \frac{l - a}{d} + 1 \]

Substituting the values:
\[ n = \frac{192 - 108}{12} + 1 \]
\[ n = \frac{84}{12} + 1 \]
\[ n = 7 + 1 = 8 \]

### Conclusion
Thus, the total numbers between 100 and 200 that are divisible by 12 is **8**.