Find the total numbers from 100 to 200, which are neither divisible by 3 nor by 5.

Answer

To find the total numbers from 100 to 200 that are neither divisible by 3 nor by 5, we can follow these steps:

### Step 1: Determine the total numbers from 100 to 200
The range from 100 to 200 includes all integers from 100 to 200. The total count of numbers in this range can be calculated as:
\[ \text{Total numbers} = 200 - 100 + 1 = 101 \]

### Step 2: Count the numbers divisible by 3
To find how many numbers from 100 to 200 are divisible by 3, we first find the largest number in this range that is divisible by 3:
- The largest number ≤ 200 that is divisible by 3 is 198.
- The smallest number ≥ 100 that is divisible by 3 is 102.

Now, we can find how many multiples of 3 are there from 102 to 198:
- The sequence of numbers divisible by 3 can be expressed as:
\[ 3n \quad \text{where} \quad n \text{ is an integer} \]
To find the values of \( n \):
- For \( 102 = 3n \) → \( n = 34 \)
- For \( 198 = 3n \) → \( n = 66 \)

Now, we calculate the count:
\[ \text{Count of multiples of 3} = 66 - 34 + 1 = 33 \]

### Step 3: Count the numbers divisible by 5
Next, we find how many numbers from 100 to 200 are divisible by 5:
- The largest number ≤ 200 that is divisible by 5 is 200.
- The smallest number ≥ 100 that is divisible by 5 is 100.

Now, we can find how many multiples of 5 are there from 100 to 200:
- The sequence of numbers divisible by 5 can be expressed as:
\[ 5m \quad \text{where} \quad m \text{ is an integer} \]
To find the values of \( m \):
- For \( 100 = 5m \) → \( m = 20 \)
- For \( 200 = 5m \) → \( m = 40 \)

Now, we calculate the count:
\[ \text{Count of multiples of 5} = 40 - 20 + 1 = 21 \]

### Step 4: Count the numbers divisible by both 3 and 5 (i.e., 15)
Now we need to find how many numbers from 100 to 200 are divisible by both 3 and 5, which means they are divisible by 15:
- The largest number ≤ 200 that is divisible by 15 is 195.
- The smallest number ≥ 100 that is divisible by 15 is 105.

Now, we can find how many multiples of 15 are there from 105 to 195:
- The sequence of numbers divisible by 15 can be expressed as:
\[ 15k \quad \text{where} \quad k \text{ is an integer} \]
To find the values of \( k \):
- For \( 105 = 15k \) → \( k = 7 \)
- For \( 195 = 15k \) → \( k = 13 \)

Now, we calculate the count:
\[ \text{Count of multiples of 15} = 13 - 7 + 1 = 7 \]

### Step 5: Apply the principle of inclusion-exclusion
Now, we can find the total count of numbers divisible by either 3 or 5:
\[ \text{Count of numbers divisible by 3 or 5} = (\text{Count of multiples of 3}) + (\text{Count of multiples of 5}) - (\text{Count of multiples of 15}) \]
\[ = 33 + 21 - 7 = 47 \]

### Step 6: Calculate the numbers neither divisible by 3 nor by 5
Finally, we subtract the count of numbers divisible by either 3 or 5 from the total count of numbers from 100 to 200:
\[ \text{Count of numbers neither divisible by 3 nor by 5} = \text{Total numbers} - \text{Count of numbers divisible by 3 or 5} \]
\[ = 101 - 47 = 54 \]

### Final Answer
The total numbers from 100 to 200 that are neither divisible by 3 nor by 5 is **54**.

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