How many numbers between 1 and 1000 are divisible by 7?

Options

A. 777
B. 142
C. 143
D. None of these

Answer

To find how many numbers between 1 and 1000 are divisible by 7, we can follow these steps:

### Step 1: Identify the first and last numbers divisible by 7
The first number greater than or equal to 1 that is divisible by 7 is 7 itself. The largest number less than or equal to 1000 that is divisible by 7 can be found by dividing 1000 by 7 and taking the integer part of the result.

### Step 2: Calculate the largest number divisible by 7
To find the largest number divisible by 7, we perform the following calculation:
\[ \text{Largest number} = 7 \times \left\lfloor \frac{1000}{7} \right\rfloor \]
Calculating \( \frac{1000}{7} \):
\[ \frac{1000}{7} \approx 142.857 \]
Taking the floor value:
\[ \left\lfloor 142.857 \right\rfloor = 142 \]
Now, multiply by 7:
\[ 7 \times 142 = 994 \]
So, the largest number divisible by 7 between 1 and 1000 is 994.

### Step 3: Use the formula for the number of terms in an arithmetic sequence
The numbers divisible by 7 form an arithmetic sequence where:
- The first term \( a = 7 \)
- The common difference \( d = 7 \)
- The last term \( l = 994 \)

The formula for the \( n \)-th term of an arithmetic sequence is:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 994 = 7 + (n - 1) \cdot 7 \]

### Step 4: Solve for \( n \)
Rearranging the equation:
\[ 994 - 7 = (n - 1) \cdot 7 \]
\[ 987 = (n - 1) \cdot 7 \]
Now, divide both sides by 7:
\[ n - 1 = \frac{987}{7} \]
Calculating \( \frac{987}{7} \):
\[ \frac{987}{7} = 141 \]
So,
\[ n - 1 = 141 \]
Adding 1 to both sides gives:
\[ n = 142 \]

### Conclusion
Thus, the total number of numbers between 1 and 1000 that are divisible by 7 is **142**.

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