Divisibility Rules for 7 , 11 and 13

A guest post by Dr. Cecily Zacharias from Oklahoma city, The United Sates of America. Currently she is an instructor at Oklahoma City Community College

Rules for divisibility of 7 , 11 and 13

It is equally good for 11 and 13.

Step 1    Divide into groups of three from the right.                                                      245782          245  782

______________________________________________________________-1           1
Step 2.   Write 1,-1,1,-1(alternate 1 and -1) in a row above the number          245       782
( start at the right end  and go left)

____________________________________________________________-1            1
Step 3.   Divide each Group by 7 (or 11 or 13, whatever the divisor is )     245         782
0             5

* You can avoid step 2, by simply subtracting first remainder from the second. In this case it will be simply, 5 – 0 = 5
Step 4.  Multiply the corresponding numbers in the top row and bottom row and add     0x -1  +  5x 1   = 5

** Step 4 can also be avoided.
Step 5. a.  If the sum obtained is zero, The number is divisible by 7 (or 11 or 13 )
b. If the sum is positive, then that is the remainder when we divide the number by 7 (  or 11 or 13 )
c. If the sum  is negative, then add 7 (or 11 or 13 )  to get the remainder.
The sum is always less than the divisor.

In the example given, the sum is 5 . Which can be verified.
When 245782 is divided by 7 by long division, the quotient is 35111 and remainder is 5.

If we test for divisor 11, the bottom row will be    3      1
The sum of products of the two rows is -2. since it is negative , add 11 .
So the remainder will be 9
Actual division gives the quotient to be 22343 and remainder 9.

It is the same  method for dividing by 13 too.

If you want you can simplify the  steps 2 ,to 4 as
Find the remainders in each group and alternately add and subtract the remainders starting from  the right. Then use step 5.

On behalf of all the QuickerMaths.com users, I  am highly grateful for her contribution.

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