Correct Option (3)
Let the two natural numbers be $n_1$ and $n_2$, such that $n_2 - n_1 = 10$. We are interested in the count of natural numbers divisible by 5 that lie strictly between $n_1$ and $n_2$. These numbers are $n_1+1, n_1+2, \dots, n_2-1$. This sequence comprises 9 consecutive natural numbers.
Consider the following cases:
- Case 1: When $n_1$ is a multiple of 5.
Let $n_1 = 5k$ for some natural number $k$. Then $n_2 = 5k + 10$. The numbers strictly between $n_1$ and $n_2$ are $5k+1, 5k+2, \dots, 5k+9$. In this sequence, only $5k+5$ is a multiple of 5. For example, if $n_1=5$ and $n_2=15$, the numbers between them are $6, 7, \dots, 14$. Only $10$ is divisible by 5, which is one number.
- Case 2: When $n_1$ is not a multiple of 5.
Let $n_1 = 5k+r$, where $r \in \{1, 2, 3, 4\}$. Then $n_2 = 5k+r+10$. The numbers strictly between $n_1$ and $n_2$ are $5k+r+1, \dots, 5k+r+9$. Within any sequence of 9 consecutive natural numbers, if the sequence does not start or end with a multiple of 5, it will contain two multiples of 5.
- For example, if $n_1=1$ and $n_2=11$, the numbers between them are $2, 3, \dots, 10$. The numbers divisible by 5 are $5$ and $10$, which are two numbers.
- If $n_1=6$ and $n_2=16$, the numbers between them are $7, 8, \dots, 15$. The numbers divisible by 5 are $10$ and $15$, which are two numbers.
Since there are instances where two such numbers exist, it is possible for there to be more than one number divisible by 5 between the two given natural numbers.
Incorrect Options:
Options (1), (2), and (4) are incorrect because the number of natural numbers divisible by 5 between the two given numbers is not constant. As demonstrated, there are scenarios where only one such number exists (e.g., between 5 and 15) and scenarios where two such numbers exist (e.g., between 1 and 11). Therefore, stating there is always only one, always only two, or never any such number is inaccurate.
