The number of integers between 100 and 1000 that are not divisible by 7 is
(i) 128
(ii) 899
(iii) 771
(iv) 772

Options

A. 128
B. 899
C. 771
D. 772

Answer

To solve the problem of finding the number of integers between 100 and 1000 that are not divisible by 7, we can follow these steps:

### Step 1: Determine the range of integers
The integers we are considering are from 100 to 999 (inclusive).

### Step 2: Count the total integers in the range
To find the total number of integers between 100 and 999:
- The first integer is 100.
- The last integer is 999.
- Total integers = Last integer - First integer + 1 = 999 - 100 + 1 = 900.

### Step 3: Find the first integer divisible by 7
To find the first integer greater than or equal to 100 that is divisible by 7:
- Divide 100 by 7: \( 100 \div 7 = 14.2857 \) (approximately).
- The smallest integer greater than this is 15.
- Therefore, the first integer divisible by 7 is \( 7 \times 15 = 105 \).

### Step 4: Find the largest integer divisible by 7
To find the largest integer less than or equal to 999 that is divisible by 7:
- Divide 999 by 7: \( 999 \div 7 = 142.7143 \) (approximately).
- The largest integer less than this is 142.
- Therefore, the largest integer divisible by 7 is \( 7 \times 142 = 994 \).

### Step 5: Count the integers divisible by 7
Now, we need to find how many integers are divisible by 7 between 105 and 994:
- The integers divisible by 7 form an arithmetic sequence where:
- First term \( a = 105 \)
- Last term \( l = 994 \)
- Common difference \( d = 7 \)

To find the number of terms \( n \) in this sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n-1)d \]
Rearranging gives:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{994 - 105}{7} + 1 \]
\[ n = \frac{889}{7} + 1 \]
\[ n = 127 + 1 = 128 \]

### Step 6: Calculate the number of integers not divisible by 7
Now, we subtract the number of integers divisible by 7 from the total number of integers:
- Total integers = 900
- Integers divisible by 7 = 128
- Therefore, integers not divisible by 7 = \( 900 - 128 = 772 \).

### Final Answer
The number of integers between 100 and 1000 that are not divisible by 7 is **772**.