"Sum of all 3 digit numbers divisible by 8" is a difficult problem having had by the students who study math to get prepared for competitive exams.

For some students, getting answer for the questions like "Find the sum of all 3 digit numbers divisible by 8" is never being easy and always it is a challenging one.

Once we know the concept and method of solving, solving the above problem will not be a challenging one.

To get the sum of 3 digit numbers divisible by 8, first we have to find the first and last 3 digit numbers divisible by 8.

The smallest 3 digit number = 100

The first 3 digit number is also 100.

To find the first 3 digit number divisible by 8, we divide the very first 3 digit number 100 by 8.

100/8 = 12.5

We have decimal in the result of 100/8.

Clearly the first 3 digit number 100 is not exactly divisible by 8.

Let us divide the second 3 digit number 101 by 8.

101/8 = 12.625

We have decimal in the result of 101/8 also.

So, the second 3 digit number 101 is also not exactly divisible by 8

Here, students may have some questions on the above process.

They are,

**1. Do we have to divide the 3 digit numbers by 8 starting from 100 until we get a 3 digit number which is exactly divisible by 8 ?**

**2. Will it not take a long process?**

**3. Is there any shortcut instead of dividing the 3 digit numbers 100, 101, 102.... one by one?**

There is only one answer for all the above three questions.

**That is, there is a shortcut to find the first three digit number which is exactly divisible by 8. **

**SHORTCUT**

**What has been done in the above shortcut?**

The process which has been done in the above shortcut has been explained clearly in the following steps.

**Step 1 :**

To get the first 3 digit number divisible by 8, we have to take the very first 3 digit number 100 and divide it by 8.

**Step 2 :**

When we divide 100 by 8 using long division as given above, we get the remainder 4.

**Step 3 :**

Now, the remainder 4 has to be subtracted from the divisor 8.

When we subtract the remainder 4 from the divisor 8, we get the result 4 (That is 8 - 4 = 4).

**Step 4 :**

Now, the result 4 in step 3 to be added to the dividend 100.

When we add 4 to 100, we get 104.

Now, the process is over.

**So, 104 is the first 3 digit number exactly divisible by 8. **

This is how we have to find the first 3 digit number exactly divisible by 8.

**Important Note: **

**This method is not only applicable to find the first 3 digit number exactly divisible by 8. It can be applied to find the first 3 digit number exactly divisible by any number, say "k"**

The largest 3 digit number = 999

The last 3 digit number is also 999

To find the last 3 digit number divisible by 8, we divide the very last 3 digit number 999 by 8.

999/8 = 124.875

We have decimal in the result of 999/8.

Clearly the last 3 digit number 999 is not exactly divisible by 8.

Let us divide the preceding 3 digit number 998 by 8.

998/8 = 124.75

We have decimal in the result of 998/8 also.

So, the preceding 3 digit number 998 also is not exactly divisible by 8

Here, students may have some questions on the above process.

They are,

**1.
Do we have to divide the 3 digit numbers .......997, 998, 999 by 8 until
we get a 3 digit number which is exactly divisible by 8 ?**

**2. Will it not take a long process?**

**3. Is there any shortcut instead of dividing the 3 digit numbers ...........997, 998, 999 one by one?**

There is only one answer for all the above three questions.

**That is, there is a shortcut to find the last three digit number which is exactly divisible by 8. **

**SHORTCUT**

**What has been done in the above shortcut?**

The process which has been done in the above shortcut has been explained clearly in the following steps.

**Step 1 :**

To get the last 3 digit number divisible by 8, we have to take the very last 3 digit number 999 and divide it by 8.

**Step 2 :**

When we divide 999 by 8 using long division as given above, we get the remainder 7.

**Step 3 :**

Now, the remainder 7 has to be subtracted from the dividend 999.

When we subtract the remainder 7 from the dividend 999, we get the result 992 (That is 999 - 7 = 992).

Now, the process is over.

**So, 992 is the last 3 digit number exactly divisible by 8. **

This is how we have to find the last 3 digit number exactly divisible by 8.

**Important Note: **

**The process of finding the first 3 digit number exactly divisible by 8 and the process of finding the last 3 digit number exactly divisible by 8 are completely different. **

**Be careful! Both are not same.**

**The
methods explained above are not only applicable to find the first 3 digit number and last 3 digit number exactly
divisible by 8. They can be applied to find the first 3 digit number
and last 3 digit number exactly divisible by any number, say "k"**

Let us see how to find the sum of all 3 digit numbers divisible by 8 in the following steps.

**Step 1 :**

The first 3 digit number divisible by 8 is 104.

After 104, to find the next 3 digit number divisible by 8, we have to add 8 to 104. So the second 3 digit number divisible by 8 is 112.

In this way, to get the succeeding 3 digit numbers divisible by 8, we just have to add 8 as given below.

**104, 112, 120, 128,...............................................992**

Clearly, the above sequence of 3 digit numbers divisible by 8 forms an Arithmetic Progression.

And our aim is to find the sum of the terms in the above A.P

**Step 2 :**

In the A.P 104, 112, 120, .......................................................992 , we have

first term = 104, common difference = 8, last term = 992

That is, **a = 104**, **d = 8** and **l = 992**

**Step 3 :**

The formula to find the numbers of terms in an A.P

** n = [(l-a)/d] + 1**

Plugging a = 104, l = 992 and d = 8

n = [(992-104)/8]+1

n = [888/8]+1 = 111+1

n = 112

So, number of 3 digit numbers divisible by 8 is **112**

**Step 4 :**

The formula to find the sum of "n" terms in an A.P is

= **n/2{a+l}**

Plugging a = 104, d = 8, l = 992 and n = 112, we get

= 112/2{104+992}

= 56x1096

= **61376**

**Hence, the sum of all 3 digit numbers divisible by 8 is 61376**

**The method explained above is not only applicable to find the sum of all 3 digit numbers divisible by 8. This same method can be applied to find sum of all 3 digit numbers divisible by any number, say "k".**

When students have the questions like "Find the sum of all 3 digit numbers divisible by 8", in competitive exams, they are stumbling a lot to solve. If we know the way of solving, getting answer for the questions like "Find the sum of all 3 digit numbers divisible by 8" is not a difficult task.

We hope, after having seen the methods and steps explained above, students will not find it difficult to answer the questions like "Find the sum of all 3 digit numbers divisible by 8".

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