How many numbers are there from 700 to 950 (including both) which are neither divisible by 3 nor by 7 ?

Options

A. 107
B. 141
C. 144
D. 145

Answer

To find how many numbers from 700 to 950 (inclusive) are neither divisible by 3 nor by 7, we can follow these steps:

### Step 1: Determine the total numbers from 700 to 950
The total count of numbers from 700 to 950 can be calculated as follows:

\[ \text{Total Numbers} = 950 - 700 + 1 = 251 \]

### Step 2: Count numbers divisible by 3
To find how many numbers between 700 and 950 are divisible by 3, we first find the smallest and largest multiples of 3 in this range.

- The smallest multiple of 3 greater than or equal to 700 is:
\[ 3 \times \lceil \frac{700}{3} \rceil = 3 \times 234 = 702 \]

- The largest multiple of 3 less than or equal to 950 is:
\[ 3 \times \lfloor \frac{950}{3} \rfloor = 3 \times 316 = 948 \]

Now, we can find the count of multiples of 3 from 702 to 948:
\[ \text{Count of multiples of 3} = \left(\frac{948 - 702}{3}\right) + 1 = \left(\frac{246}{3}\right) + 1 = 82 + 1 = 83 \]

### Step 3: Count numbers divisible by 7
Next, we find how many numbers between 700 and 950 are divisible by 7.

- The smallest multiple of 7 greater than or equal to 700 is:
\[ 7 \times \lceil \frac{700}{7} \rceil = 7 \times 100 = 700 \]

- The largest multiple of 7 less than or equal to 950 is:
\[ 7 \times \lfloor \frac{950}{7} \rfloor = 7 \times 135 = 945 \]

Now, we can find the count of multiples of 7 from 700 to 945:
\[ \text{Count of multiples of 7} = \left(\frac{945 - 700}{7}\right) + 1 = \left(\frac{245}{7}\right) + 1 = 35 + 1 = 36 \]

### Step 4: Count numbers divisible by both 3 and 7 (i.e., divisible by 21)
Now we need to find how many numbers are divisible by both 3 and 7, which is equivalent to finding numbers divisible by 21.

- The smallest multiple of 21 greater than or equal to 700 is:
\[ 21 \times \lceil \frac{700}{21} \rceil = 21 \times 34 = 714 \]

- The largest multiple of 21 less than or equal to 950 is:
\[ 21 \times \lfloor \frac{950}{21} \rfloor = 21 \times 45 = 945 \]

Now, we can find the count of multiples of 21 from 714 to 945:
\[ \text{Count of multiples of 21} = \left(\frac{945 - 714}{21}\right) + 1 = \left(\frac{231}{21}\right) + 1 = 11 + 1 = 12 \]

### Step 5: Apply the principle of inclusion-exclusion
To find the total count of numbers that are divisible by either 3 or 7, we use the inclusion-exclusion principle:
\[ \text{Count of numbers divisible by 3 or 7} = \text{Count of multiples of 3} + \text{Count of multiples of 7} - \text{Count of multiples of 21} \]
\[ = 83 + 36 - 12 = 107 \]

### Step 6: Find numbers neither divisible by 3 nor by 7
Finally, we subtract the count of numbers divisible by 3 or 7 from the total count:
\[ \text{Count of numbers neither divisible by 3 nor 7} = \text{Total Numbers} - \text{Count of numbers divisible by 3 or 7} \]
\[ = 251 - 107 = 144 \]

Thus, the total number of numbers from 700 to 950 that are neither divisible by 3 nor by 7 is **144**.