What is the divisibility rule for 11, 12, and 13?

Answer 1

Please see below.

Divisibility Rule for #11#
Divide the alternate digits in two different groups. Take the sum of alternate digits separately and find the difference of the two numbers. If the difference is #0# or is divisible #11#, the number is divisible by #11#.
Example: #86456293# is divided into two groups #{8,4,6,9}# and #{6,5,2,3}#. Sum of the groups is #27# and #16#, whose difference is #11# and the it is divisible by #11#, #86456293# is divisible by #11#.
Divisibility Rule for #12#
If the number is divisible by both #3# and #4#, the number is divisible by #12#. Divisibility rule of #3# is tat sum of digits is divisible by #3# and divisibility rule of #4# is that last two digits are divisible by #4#.
Example: In #185176368# sum of all the digits is #45# and is divisible by #3# and also last two digits #68# are divisible by #4#. As such the number #185176368# is divisible by #12#.
Divisibility Rule for #13#
Recall the divisibility rule of #7#, this works for #13# too.

Mark off the digits in groups of three starting from the right (just like we do when we put commas in large numbers).

Now add up alternate group of numbers and find the difference between the two. If the difference is divisible by #13#, entire number is divisible by #13#.
For example #123448789113#, these are grouped as #123#, #448#, #789# and #113#
and #123+789=912# and #448+113=561#.
As difference between #912-561=351#
As #351# is divisible by #13#, #123448789113# is divisible by #13#
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Answer 2

The divisibility rule for:

  1. 11: Add the digits in odd places and then the digits in even places. If the difference between the two sums is divisible by 11, then the whole number is divisible by 11.

  2. 12: A number is divisible by 12 if it is divisible by both 3 and 4. To check divisibility by 3, add up the digits of the number. If the sum is divisible by 3, then the number is divisible by 3. To check divisibility by 4, examine the last two digits of the number. If the number formed by the last two digits is divisible by 4, then the whole number is divisible by 4.

  3. 13: There isn't a simple and widely known rule for divisibility by 13. Divisibility by 13 is usually checked by division.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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