How Many 3-Digit Natural Numbers are Divisible by 9

Edited By Team Careers360 | Updated on Apr 27, 2023 02:08 PM IST

Introduction

Three-digit numbers begin at 100 and terminate at 999. As a result, the lowest three-digit number is 100, while the greatest three-digit number is 999. In mathematics, the number is defined by the number of digits. The place values of a three-digit number will be used to describe it. Natural numbers are the numbers used in mathematics for counting and ranking. Cardinal numbers are those used for counting, while ordinal numbers are those used for ranking.

1. Solution

The three-digit numbers divisible by 9 are 108, 117, 126,…., 999. Clearly, these numbers are in AP. Here. a= 108 and

\begin{equation}

d= 117-108

\Rightarrow 9

\end{equation}

Let this AP contains n terms. Then.

an = 999

\begin{equation}

[an = a + (n -1)d]

\end{equation}

The Arithmetic Progression is the most often used sequence in mathematics, using simple formulae. Arithmetic Progression (AP) is a mathematical series in which the difference between two consecutive terms is always a constant.

\begin{equation}

\Rightarrow 9n +99 =999

\Rightarrow 9n = 999-99 = 900

\Rightarrow n=100

\end{equation}

2. List of numbers

Numbers	Reasons (Remainders=0)
\begin{equation} 108/9= 12 \end{equation}	108 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 108/9 = 12, and 0 is the remainder, 108 is divisible by 9 (we say "108 divided by 9").
\begin{equation} 117/9= 13 \end{equation}	117 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 117/9 = 13, and 0 is the remainder, 117 is divisible by 9 (we say "117 divided by 9").
\begin{equation} 126/9= 14 \end{equation}	126 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 126/9 = 14, and 0 is the remainder, 126 is divisible by 9 (we say "126 divided by 9").
\begin{equation} 135/9= 15 \end{equation}	135 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 135/9 = 15, and 0 is the remainder, 135 is divisible by 9 (we say "135 divided by 9").
\begin{equation} 144/9= 16 \end{equation}	144 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 144 /9 = 16, and 0 is the remainder, 144 is divisible by 9 (we say "144 divided by 9").
\begin{equation} 153/9= 17 \end{equation}	153 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 153 /9 = 17, and 0 is the remainder, 153 is divisible by 9 (we say "153 divided by 9").
\begin{equation} 162/9= 18 \end{equation}	162 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 162 /9 = 18, and 0 is the remainder, 162 is divisible by 9 (we say "162 divided by 9").
\begin{equation} 171/9= 19 \end{equation}	171 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 171 /9 = 19, and 0 is the remainder, 171 is divisible by 9 (we say "171 divided by 9").
\begin{equation} 180/9= 20 \end{equation}	180 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 180 /9 = 20, and 0 is the remainder, 180 is divisible by 9 (we say "180 divided by 9").
\begin{equation} 189/9= 21 \end{equation}	189 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 189/9 = 21, and 0 is the remainder, 189 is divisible by 9 (we say "189 divided by 9").
\begin{equation} 198/9= 22 \end{equation}	198 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 198/9 = 22, and 0 is the remainder, 198 is divisible by 9 (we say "198 divided by 9").
\begin{equation} 207/9= 23 \end{equation}	207 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 207/9 = 23, and 0 is the remainder, 207 is divisible by 9 (we say "207 divided by 9").
\begin{equation} 216/9= 24 \end{equation}	216 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 216/9 = 24, and 0 is the remainder, 216 is divisible by 9 (we say "216 divided by 9").
\begin{equation} 225/9= 25 \end{equation}	225 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 225/9 = 25, and 0 is the remainder, 225 is divisible by 9 (we say "225 divided by 9").
\begin{equation} 234/9= 26 \end{equation}	234 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 234/9 = 26, and 0 is the remainder, 234 is divisible by 9 (we say "234 divided by 9").
\begin{equation} 243/9= 27 \end{equation}	243 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 243/9 = 27, and 0 is the remainder, 243 is divisible by 9 (we say "243 divided by 9").
\begin{equation} 252/9= 28 \end{equation}	252 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 252 /9 = 28, and 0 is the remainder, 252 is divisible by 9 (we say "252 divided by 9").
\begin{equation} 261/9= 29 \end{equation}	261 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 261/9 = 29, and 0 is the remainder, 261 is divisible by 9 (we say "261 divided by 9").
\begin{equation} 270/9= 30 \end{equation}	270 is divisible by 9, so it can be divided equally by that number; that is, the division gives a whole integer. Because 270 /9 = 20, and 0 is the remainder, 270 is divisible by 9 (we say "270 divided by 9").

3. Notes

We applied the A.P. idea. To answer the provided question, an arithmetic progression is a series in which the difference between every two subsequent integers is a constant. We must recall the nth formula. An A.P.'s phrase, "The usage of n is a common mistake,"

instead of

\begin{equation}

n-1

\end{equation}