Question

The total number of Natural numbers that lie between 10 and 300 and are divisible by 9 is

• A. 32
• B. 30
• C. 33
• D. 34

Answer 1

The Correct Option is A

To solve the problem of finding the total number of natural numbers between 10 and 300 that are divisible by 9, we can use the concept of arithmetic sequences. A number is divisible by 9 if it can be expressed as \(9k\) for some integer \(k\).

First, find the smallest multiple of 9 greater than 10. We start by calculating: 

\[\lceil \frac{10}{9} \rceil = \lceil 1.11 \rceil = 2\]

So, the first number is \(9 \times 2 = 18\).

Next, find the largest multiple of 9 less than 300:

\[\lfloor \frac{300}{9} \rfloor = \lfloor 33.33 \rfloor = 33\]

Thus, the last number is \(9 \times 33 = 297\).

The natural numbers divisible by 9 between 10 and 300 form an arithmetic sequence with the first term \(a = 18\) and the last term \(l = 297\), where the common difference \(d = 9\). The general form for the terms of this sequence can be defined as:

\[a_n = a + (n-1)d = 18 + (n-1) \times 9\]

Setting \(a_n = 297\), we can solve for \(n\):

\[18 + (n-1) \times 9 = 297\]

Solving for \(n\), we get:

\[9(n-1) = 297 - 18\]

\[9n - 9 = 279\]

\[9n = 288\]

\[n = \frac{288}{9} = 32\]

Therefore, there are 32 natural numbers between 10 and 300 that are divisible by 9.

Natural numbers count:

32