Total number of integers between 100 and 200 which are divisible by both 2 and 3 is

Options

A. (a) 9
B. (b) 13
C. (c) 15
D. d) 17

Answer

To find the total number of integers between 100 and 200 that are divisible by both 2 and 3, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
The first step is to find the LCM of 2 and 3. The LCM of 2 and 3 is 6. This means any number that is divisible by both 2 and 3 must also be divisible by 6.

### Step 2: Identify the Range
We are looking for integers between 100 and 200. Therefore, we need to find the multiples of 6 that fall within this range.

### Step 3: Find the First Multiple of 6 Greater than 100
To find the first multiple of 6 that is greater than 100, we can divide 100 by 6 and round up to the nearest whole number:
\[ 100 \div 6 \approx 16.67 \]
Rounding up gives us 17. Now, we multiply 17 by 6 to find the first multiple:
\[ 17 \times 6 = 102 \]
So, the first multiple of 6 greater than 100 is 102.

### Step 4: Find the Last Multiple of 6 Less than 200
Next, we need to find the largest multiple of 6 that is less than 200. We divide 200 by 6:
\[ 200 \div 6 \approx 33.33 \]
Rounding down gives us 33. Now, we multiply 33 by 6 to find the last multiple:
\[ 33 \times 6 = 198 \]
So, the last multiple of 6 less than 200 is 198.

### Step 5: Count the Multiples of 6 from 102 to 198
Now, we need to count how many multiples of 6 are there from 102 to 198. The multiples of 6 can be represented as:
\[ 6n \text{ where } n \text{ is an integer} \]
We have:
- The first multiple (n=17) is 102.
- The last multiple (n=33) is 198.

To find the total number of multiples, we can calculate:
\[ n = 33 - 17 + 1 = 17 \]
So, there are 17 integers between 100 and 200 that are divisible by both 2 and 3.

### Final Answer
The total number of integers between 100 and 200 that are divisible by both 2 and 3 is **17**.

---