DIVISIBILITY RULE FOR 7

By Gaurav
In Quant
Aug 12th, 2008
8 Comments
856 Views

A whole number, N, is a multiple of 7 if the following procedure leads to
another multiple of 7*:Divisibility rule for 7
(1) Subtract the ones digit from N,
(2) Dividing the result by 10, and
(3) Subtract—from that result—twice the original ones digit.
*the multiple of 7 may be 0 or negative.
This is best explained in an example. Consider 7 x 39 = 273, so 273 is a multiple of 7. Applying the
Divisibility Rule for 7 to 273 yields the following:

(1) Subtract the ones digit from N: Subtract 3 from 273: 273
– 3
270
(2) Dividing the result by 10: Divide 270 by 10: 270 ÷ 10 = 27
(3) Subtract—from that result—twice
the original ones digit: Subtract, from 27, twice 3:

27

– 6

21:  a multiple of 7
Since this procedure leads to a multiple of 7, it must be that the original number, 273, is a
multiple of 7, as shown in this long division:
A much quicker way to use the procedure is shown here
2 7 3
– 6
2 1
x 2
Since 21 is a multiple of 7, it must
be that 273 is a multiple of 7.

Let’s apply this more direct procedure to a larger multiple of 7: 7 x 568 = 3,976:
3 9 7 6
– 1 2
3 8 5
x 2
We don’t know if 385 is a multiple of 7
or not, so we continue the procedure.
3 8 5
– 1 0
2 8

Since 28 is a multiple of 7, it must be that
both 385 and 3,976 are multiples of 7.
What is the proof behind this procedure?
Consider, without loss of generality, a three digit number, N = 100a + 10b + c, where a, b, and c are
single digits, a ≠ 0.
Applying the Divisibility Rule for 7 to 100a + 10b + c, we get:
(1) Subtract c: 100a + 10b
(2) Divide by 10: 10a + b
(3) Subtract twice c: 10a + b – 2c
Claim: 100a + 10b + c is a multiple of 7 if and only if 10a + b – 2c is a multiple of 7
P

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8 Responses to “DIVISIBILITY RULE FOR 7”

  1. Ankit says:

    This is a nice method but not feasible in exams….
    just think doing all this or just divide the no and find it……..

    also 7 is a small number so direct division is not so difficult

    what u say tutes??

  2. gaurav says:

    In exams, the thing which matters is the comfort level. If you are comfortable with this method then subconsciously you will use this method and if not then you will directly divide the number.

    So overall this method is an add-on for those who are not comfortable with direct division.

  3. nicra says:

    Hmmm… confusing. I wouldn’t understand it. direct division is the only way it seems.

  4. vaibhav says:

    nice method. even seconds matter in cat. there is no harm using such tweaks. but these should be practiced thoroughly. If you learn it today and apply it directly on Cat day without any practice then you are bound to fail.

  5. Qi Xuan says:

    ok, not so feasible but kind of likable!

  6. Bhanu says:

    It is soo confusing.I can`t understand any thing

  7. Abhishek says:

    The people who found this method confusing..should try it again.
    It’s very gud after a little practise.

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