To get the sum of 3 digit numbers divisible by 6, first we have to find the first and last 3 digit numbers divisible by 6.
The first and also the smallest 3 digit number is 100.
To find the first 3 digit number divisible by 6, we have to divide the very first 3 digit number 100 by 6
100/6 = 16.67
We have decimal in the result of 100/6.
Clearly the first 3 digit number 100 is not exactly divisible by 6
Let us divide the second 3 digit number 101 by 6
101/6 = 16.83
We have decimal in the result of 101/6 also.
So, the second 3 digit number 101 is also not exactly divisible by 6
Here, students may have some questions on the above process.
They are,
1. Do we have to divide the 3 digit numbers by 6 starting from 100 until we get a 3 digit number which is exactly divisible by 6 ?
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 6.
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 6, we have to take the very first 3 digit number 100 and divide it by 6.
Step 2 :
When we divide 100 by 6 using long division as given above, we get the remainder 4.
Step 3 :
Now, the remainder 4 has to be subtracted from the divisor 6.
When we subtract the remainder 4 from the divisor 6, we get the result 2 (That is 6 - 4 = 2).
Step 4 :
Now, the result 2 in step 3 to be added to the dividend 100.
When we add 2 to 100, we get 102
Now, the process is over.
So, 102 is the first 3 digit number exactly divisible by 6
This is how we have to find the first 3 digit number exactly divisible by 6
Important Note :
This method is not only applicable to find the first 3 digit number exactly divisible by 6. It can be applied to find the first 3 digit number exactly divisible by any number, say k.
The last and also the largest 3 digit number is 999.
To find the last 3 digit number divisible by 6, we divide the very last 3 digit number 999 by 6.
999/6 = 166.5
We have decimal in the result of 999/6.
Clearly the last 3 digit number 999 is not exactly divisible by 6.
Let us divide the preceding 3 digit number 998 by 6.
998/6 = 166.33
We have decimal in the result of 998/6 also.
So, the preceding 3 digit number 998 also is not exactly divisible by 6
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 6 until we get a 3 digit number which is exactly divisible by 6 ?
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 6.
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 6, we have to take the very last 3 digit number 999 and divide it by 6.
Step 2 :
When we divide 999 by 6 using long division as given above, we get the remainder 3.
Step 3 :
Now, the remainder 3 has to be subtracted from the dividend 999.
When we subtract the remainder 3 from the dividend 999, we get the result 996 (That is 999 - 3 = 996).
Now, the process is over.
So, 996 is the last 3 digit number exactly divisible by 6.
This is how we have to find the last 3 digit number exactly divisible by 6.
Important Note :
The process of finding the first 3 digit number exactly divisible by 6 and the process of finding the last 3 digit number exactly divisible by 6 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 6. 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 6 in the following steps.
Step 1 :
The first 3 digit number divisible by 6 is 102.
After 102, to find the next 3 digit number divisible by 6, we have to add 6 to 102. So the second 3 digit number divisible by 6 is 108.
In this way, to get the succeeding 3 digit numbers divisible by 6, we just have to add 6 as given below.
102, 108, 114, 120,.......................996
Clearly, the above sequence of 3 digit numbers divisible by 6 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
102, 108, 114, 120,.......................996
we have
first term = 102
common difference = 6
last term = 996
That is,
a = 102
d = 6
l = 996
Step 3 :
The formula to find the numbers of terms in an arithmetic sequence is given by
n = [(l - a) / d] + 1
Substitute a = 102, l = 996 and d = 6.
n = [(996 - 102) / 6] + 1
n = [894/6] + 1
n = 149 + 1
n = 150
So, number of 3 digit numbers divisible by 6 is 150.
Step 4 :
The formula to find the sum of 'n' terms in an arithmetic sequence is given by
= (n/2)(a + l)
Substitute a = 102, d = 6, l = 996 and n = 150.
= (150/2)(102 + 996)
= 75 x 1098
= 82350
So, the sum of all 3 digit numbers divisible by 6 is 82350.
Note :
The method explained above is not only applicable to find the sum of all 3 digit numbers divisible by 6. This same method can be applied to find sum of all 3 digit numbers divisible by any number, say k.
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