Rule of Divisibility by 11
Consider the number _405,592_.
In order to determine which digits are in the _odd places_, should you begin counting from the left or the right?
Great!
First, counting from the _left_:
the sum of digits in the odd (_1st, 3rd and 5th_) places = 4 + 5 + 9 = _18_.
Now, the sum of digits in the even (_2nd, 4th and 6th_) places = 0 + 5 + 2 = _7_.
The _difference_ between these two sums is 18 - 7 = _11_, which is clearly divisible by 11.
Incorrect.
This is the sum of _all_ the digits.
Now, counting from the _right_:
the sum of digits in the odd (_1st, 3rd and 5th_) places = 2 + 5 + 0 = _7_.
Now, the sum of digits in the even (_2nd, 4th and 6th_) places = 9 + 5 + 4 = _18_.
The _difference_ between these two sums is 7 - 18 = _-11_, which is _also_ divisible by 11.
So, regardless of where you start counting, the difference between the sum of its digits in the _odd_ places and the sum of its digits in the _even_ places is divisible by 11.
Therefore, the original number, 405,592 is also divisible by 11.
Incorrect.
This is the sum of the _even_ digits.
It actually doesn't matter.
As long as the difference between the sums of odd and even placed integers is divisible by 11, the original number will be divisible by 11.
For this example, count from the _left_ and determine the _sum_ of the digits in the _odd_ places.
Putting it all together, is _358,627_ divisible by 11?
Incorrect.
Correct.
Counting from right to left, (just for variety) you have:
- the sum of digits in the odd (_1st, 3rd and 5th_) places = 7 + 6 + 5 = _18_.
- the sum of digits in the even (_2nd, 4th and 6th_) places = 2 + 8 + 3 = _13_.
The _difference_ between these two sums is 18 -13 = _5_, which is not divisible by 11.
Therefore the original number _358,627 is not divisible by 11_.
__To summarize:__
[[summary]]
It actually doesn't matter.
As long as the difference between the sums of odd and even placed integers is divisible by 11, the original number will be divisible by 11.
Nice catch!
It actually doesn't matter.
As long as the difference between the sums of odd and even placed integers is divisible by 11, the original number will be divisible by 11.
The _rule of divisibility by 11_ is a little complicated, but it still can be applied easily in your head to save you time on the GRE.
A number is _divisible by 11_ if the _difference_ between the sum of its digits in the _odd_ places and the sum of its digits in the _even_ places is divisible by 11.
In mathematical formula form:
_(sum of digits in odd places) - (sum of digits in even places) = a number that is divisible by 11_.
Left
Right
Doesn't matter
7
18
25
Yes.
No.
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