Divisibility rules

How to check the

divisibility of any positive integer by 7 and 11

Now, there are certain rules; Which are used to check whether a positive integer, no matter how large it may be is divisible by 7 and 11. If you want to know more in this regard, then please continue the read below:-

1. To check the divisibility by 7:-

We use -2 as the osculator, that is:- we do repeated subtractions of the product of the units digit, with -2; from the truncated portion of the number, until finally a positive integer is obtained which could be explicitly determined that it is a multiple of seven or not.

• Example_1:- let's Let's try 52,871 as our test case:-
• Step-1:-let's separate out the units digit =1. Then multiply this 1 with minus two and add it to the truncated part of the integer that is 5287, that is:- units digit = 1 and truncated portion = 5287.
• Step-2:-Now let's do the conventional method to get the new number, that is the new number = 5287 - 2X1 =5285.
• Now let's repeat the same step to get 528-2X5 = 518 = new number.
• Now repeat again to get 51-2X8 = 35.
• And that's it we all know 35 is a multiple of seven, that is:- 35/7 = 5. And although you can repeat the previous steps in this case as well to further verify that:- 3-2X5 = -7; which is the same as 0 when speaking of remainders with division.
• Example 2:- let's consider a much easier case, since the entire process has already been

explained through a more complicated example. That is:- let's consider 119. The following are the steps:-

• New number = 11-2x9 =11-18 = -7.
• -7+7 = 0 ; (you can add any integral multiple of the divisor, to come down to a number which shall be the same as that of the original number in terms of the remainder with the divisor ).
• Result:- since the remainder, in this case, = 0, hence it could be concluded that 119 is an integral multiple of 7 and that is verified using the fact that:- 7*17 = 119.

Tell that is:- 37,800 divisible by 7 or not, by commenting in the comments section.

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To check the divisibility by 11:- in this case, what you got to do is that:- starting from the extreme right, you take the sum of the digits which occur at the odd positions and subtract from it the sum of the digits which occur at the even positions. Now, if this difference of sums comes out to be zero or an integral multiple of 11, then the given integer is divisible by 11 otherwise indivisible.

• Example_1:- let's verify whether 14,641 is an integral multiple of 11 or not.

        The following are the steps:-

• Step-1:-Take the sum of the alternating digits at the odd indices =1+6+1 = 8.
• Step-2:-Take the sum of the alternating digits at the even indices =4+4 = 8 and the sum of alternating digits at the odd indices =1+6+1 = 8.
• Step-3:-Compute the difference of sums = 8-8 = 0
• Result:- since the difference is 0 hence according to the rule14,641 is an integral multiple of 11 and that could be verified by means of the fact that 14641/11 = 1331.

Tell whether is 16105100 divisible by 11 or not, by commenting in the comments section.