# How many three-digit numbers are divisible by 7?

By Shivank Goel|Updated : May 31st, 2023

A total of 128 three-digit numbers are divisible by 7. In this problem, we aimed to count the number of three-digit numbers that can be divided by 7 without leaving a remainder. By considering the range of three-digit numbers and identifying those that are divisible by 7, we can get the answer. Read further to know how many three-digit numbers are divisible by 7.

## Divisibility of 7

The divisibility of 7 helps us determine if a given number can be evenly divided by 7 without leaving a remainder. To check the divisibility of a number by 7, we can employ a simple rule. The rule states that if the alternating sum of the digits of a number is divisible by 7, then the number itself is also divisible by 7.

### Divisibility of 7 Rule

• Start from the rightmost digit of the number and assign alternating positive and negative signs to the digits.
• Add up these signed digits.
• If the resulting sum is divisible by 7, then the original number is divisible by 7. Otherwise, it is not.

For example: let's take the number 364.

Starting from the right, we have 4 - 6 + 3 = 1.

Since 1 is not divisible by 7, 364 is not divisible by 7.

## Total Three-Digit Numbers Divisible by 7

We should find the number of three-digit numbers which are divisible by 7

The nth term of AP is aₙ = a + (n - 1)d

Where

aₙ is the nth term,

a is the first term,

d is a common difference,

n is the number of terms

First three-digit number divisible by 7 = 105

Next number = 105 + 7 = 112

So the series is 105, 112, 119, ...

It forms an AP with 105 as the first term, and 7 is the common difference

If we divide 999 by 7, 5 will be the remainder

999 - 5 = 994 is the maximum possible three-digit number, which is divisible by 7

So the final sequence is 105, 112, 119, .... 994

the nth term of an AP = 994

a = 105

d = 7

aₙ = 994

We should find n

nth term of an A.P. is aₙ = a + (n - 1)d

994 = 105 + (n - 1)7

889 = (n - 1)7

n - 1 = 889/7

n - 1 = 127

n = 127 + 1

n = 128

Therefore, when asked How many three-digit numbers are divisible by 7? then the answer will be 128 three-digit numbers are divisible by 7.

Summary:

## How many three-digit numbers are divisible by 7?

128 three-digit numbers are divisible by 7. The divisibility rule of 7 can be used to find these numbers. This rule is a helpful tool in solving mathematical problems and determining if a given number is divisible by 7.

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