How many numbers from 1 to 100 which are completely divisible by 7?

Question

Accepted Answer

To determine how many numbers from 1 to 100 are completely divisible by 7, we need to find all the multiples of 7 within this range.

1. **Identify the smallest multiple of 7 within the range:**
   - The smallest multiple of 7 that is greater than or equal to 1 is 7 itself (7 x 1 = 7).

2. **Identify the largest multiple of 7 within the range:**
   - The largest multiple of 7 that is less than or equal to 100 is 98 (7 x 14 = 98).

3. **Count the multiples of 7 between 7 and 98:**
   - The multiples of 7 from 7 to 98 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, and 98.
   - To count these, note that they form an arithmetic sequence where the first term (a) is 7, the common difference (d) is 7, and the last term (l) is 98.

4. **Use the formula for the number of terms in an arithmetic sequence:**
   - The formula for the number of terms (n) in an arithmetic sequence is given by:
     $$
     n = \frac{l - a}{d} + 1
     $$
   - Substituting the values, we get:
     $$
     n = \frac{98 - 7}{7} + 1 = \frac{91}{7} + 1 = 13 + 1 = 14
     $$

Therefore, there are 14 numbers from 1 to 100 that are completely divisible by 7, which confirms that Option 4 is correct.

**Explanation of why other options are incorrect:**

- **Option 1 (9):** This option is incorrect because, as calculated, there are more than 9 multiples of 7 within the range of 1 to 100.
  
- **Option 2 (11):** This option is incorrect because the actual count of multiples of 7 is 14, which is greater than 11.

- **Option 3 (17):** This option is incorrect because it overestimates the number of multiples of 7 within the given range. The correct count is 14, not 17.

In conclusion, the correct answer is Option 4, as there are exactly 14 numbers from 1 to 100 that are divisible by 7.