How many natural numbers are there between 23 and 100 which are exactly divisible by 6 ?

Question

Accepted Answer

To determine how many natural numbers between 23 and 100 are exactly divisible by 6, we need to find the numbers in this range that are multiples of 6.

1. **Identify the smallest multiple of 6 greater than 23:**
   - The multiples of 6 are 6, 12, 18, 24, 30, 36, and so on.
   - The smallest multiple of 6 greater than 23 is 24.

2. **Identify the largest multiple of 6 less than or equal to 100:**
   - Continuing the sequence of multiples of 6, we have 90, 96, 102, etc.
   - The largest multiple of 6 less than or equal to 100 is 96.

3. **Determine the number of multiples of 6 between 24 and 96:**
   - The sequence of multiples of 6 from 24 to 96 is: 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96.
   - To find the number of terms in this sequence, we use the formula for the nth term of an arithmetic sequence: $ a_n = a_1 + (n-1) \times d $, where $ a_1 $ is the first term, $ d $ is the common difference, and $ a_n $ is the nth term.
   - Here, $ a_1 = 24 $, $ d = 6 $, and $ a_n = 96 $.
   - Solving for $ n $: $ 96 = 24 + (n-1) \times 6 $
   - $ 96 - 24 = (n-1) \times 6 $
   - $ 72 = (n-1) \times 6 $
   - $ n-1 = 12 $
   - $ n = 13 $

Therefore, there are 13 natural numbers between 23 and 100 that are exactly divisible by 6, which supports Option 3 as the correct answer.

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

- **Option 1 (11):** This option is incorrect because, as calculated, there are 13 numbers, not 11.
- **Option 2 (12):** This option is incorrect because the sequence includes 13 numbers, not 12.
- **Option 4 (14):** This option is incorrect because the count of numbers divisible by 6 in the given range is 13, not 14.

Thus, the correct answer is indeed Option 3, as there are 13 natural numbers between 23 and 100 that are divisible by 6.