How many numbers are there between 15 to 160 which are divisible by 17 but not by 4?

Options

A. 7
B. 2
C. 6
D. 4

Explanation

To solve the problem of finding how many numbers between 15 and 160 are divisible by 17 but not by 4, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range and the First Multiple of 17:**
- The first multiple of 17 that is greater than or equal to 15 is 17 itself.

2. **Identify the Last Multiple of 17:**
- The last multiple of 17 that is less than or equal to 160 can be found by dividing 160 by 17 and taking the integer part of the result.
- \( \frac{160}{17} \approx 9.41 \) (taking the integer part gives us 9).
- Therefore, the last multiple of 17 is \( 17 \times 9 = 153 \).

3. **List the Multiples of 17:**
- The multiples of 17 between 15 and 160 are:
- 17, 34, 51, 68, 85, 102, 119, 136, 153.

4. **Count the Total Multiples of 17:**
- We can count these multiples: 17, 34, 51, 68, 85, 102, 119, 136, 153.
- There are 9 multiples of 17 in total.

5. **Identify the Multiples of 4 Among These:**
- We need to find which of these multiples of 17 are also divisible by 4.
- The least common multiple (LCM) of 17 and 4 is \( 68 \).
- The multiples of 68 in our range are:
- 68 and 136.

6. **Count the Multiples of 68:**
- The multiples of 68 between 15 and 160 are: 68, 136.
- There are 2 multiples of 68.

7. **Calculate the Final Count:**
- To find the numbers that are divisible by 17 but not by 4, we subtract the count of multiples of 68 from the count of multiples of 17.
- \( 9 - 2 = 7 \).

### Final Answer:
There are **7 numbers** between 15 and 160 that are divisible by 17 but not by 4.