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 first and the smallest 3 digit number is 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 last and the largest 3 digit number is 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 sequence.
And our aim is to find the sum of the terms in the above arithmetic sequence.
Step 2 :
In the arithmetic sequence
104, 112, 120, 128,...............................992,
we have
first term = 104
common difference = 8
last term = 992
That is,
t_{1} = 104
d = 8
ℓ = 992
Step 3 :
The formula to find the numbers of terms in an arithmetic sequence is given by
Substitute t_{1} = 104, ℓ = 992 and d = 8.
n = 111 + 1
n = 112
So, number of 3 digit numbers divisible by 8 is 112.
Step 4 :
The formula to find the sum of first n terms in an arithmetic sequence is given by
Substitute a = 104, d = 8, ℓ = 992 and n = 112.
= (¹¹²⁄₂)(104 + 992)
= 56(1096)
= 61376
So, the sum of all 3 digit numbers divisible by 8 is 61376.
Note :
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.
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