How Many 3-digit Numbers are Divisible By 7

Introduction

The divisibility rule, also known as the divisibility test, is a technique used in mathematics to determine whether a given number can be divided by a fixed divisor without actually performing the division operation. When a number is divided by a divisor, this method typically uses the digits to determine the original number. We can state that the remainder should be zero and the quotient should be a whole number if a number is perfectly divisible by another number.

In order to determine whether a number can be divided by 7 entirely without leaving a remainder, we can use the rule of divisibility of 7. To figure this out, we typically perform the division arithmetic operation. However, there is a quick way to determine whether a number is divisible by 7 using the rule of divisibility of 7. The last digit of a number is chosen, multiplied by 2, and then subtracted from the remainder of the number to its left according to the divisibility rule of 7. To ensure that the difference is entirely divisible by 7, we look to see if it is a 0 or a multiple of 7. Let's check how many 3-digit numbers there are that can be divided by 7.

Calculation

To find the number of 3-digit numbers which are divisible by 7, consider the following steps -

The nth term of AP - a_{n}= a+(n-1)d

Here, a_{n} = n^{th} term, a = first term, d = common difference and n = number of terms.

First 3-digit number divisible by 7 = 105

Second number = 105 + 7 = 112

Therefore, the series = 105, 112, 119, …….

As a result, an AP is formed by the numbers 105, 112, 119, etc., with a common difference of 7.

The remainder after multiplying 999 by 7 is 5.

Clearly, 999 - 5 = 994, the maximum possible 3-digit number that is divisible by 7.

So, the final sequence = 105, 112, 119, …., 994.

Let 994 be the n^{th} term of this AP is

a = 105

d = 7

a_{n} = 994

n = ?

So, n^{th} term of an AP is -

a_{n}= a+(n-1)d

When we substitute the values in the equation above, we obtain -

994 = 105 + (n - 1)7

889 = (n - 1)7

n - 1 = 889 / 7

n - 1 = 127

n = 127 + 1

n = 128.

Conclusion

There are 128 numbers with three digits that can be divided by seven.