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# Divisibility Rules

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Current Student
Joined: 28 Mar 2012
Posts: 304
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38

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23 Jun 2012, 01:16
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Hi,

You all must have used divisibility rules in one of Those questions. But have you ever wondered, why is it that you take the units digit as even to check the divisibility by 2, why sum of all the digits is checked for divisibility by 3...

Still wondering, then rest of the post is for you:

it can be written as 100a + 10b + c, where, a, b & c are non negative integers:

Divisibility rule for 2

$$\frac {100a + 10b + c}2$$
100a, 10b are divisible by 2, so, whether the number is divisible by 2 depends on c and that's the unit digit.

Divisibility rule for 3

$$\frac {100a + 10b + c}3$$
100a+10b+c=99a+a+9b+b+c=(99a+9b)+(a+b+c)
(99a+9b) is divisible by 3, so, the number is divisible by 3 when a+b+c is divisible by 3 and that's sum of all digit in a number.

Divisibility rule for 4

$$\frac {100a + 10b + c}4$$
100a is divisible by 4, so, whether the number is divisible by 4 depends on 10b+c ,i.e, last two digits of a number.

Divisibility rule for 5

This one is similar to divisibility rule for 2.
$$\frac {100a + 10b + c}5$$, since 100a, 10b are divisible by 5, what remains is c,
so, c has to be 0 or 5 for number to be divisible by 5.

Divisibility rule for 6

The classical rule tells us that if the number is divisible by 2 & 3 it is divisible by 6.

Divisibility rule for 7

That's a tricky one,
$$\frac {100a + 10b + c}7$$
100a+10b+c=98a+2a+7b+3b+c=(98a+7b)+(2a+3b+c)
(98a+7b) is divisible by 7, so you have to only check (2a+3b+c)
For ex - 143, 2*1+3*4+1*3=17, not divisible by 7
154 = 2*1+3*5+1*4=21, divisible by 7.
For numbers with 4 or more digits, I believe you can find out the rule now!

Divisibility rule for 8

Similar to that of 2 & 4.

Divisibility rule for 9

Similar to that of 3.

Divisibility rule for 11

$$\frac {100a+10b+c}{11}$$
100a+10b+c=99a+a+11b-b+c=(99a+11b)+(a+c-b)
(99a+11b) is divisible by 11, so we have to check (a+c-b), i.e., difference of sum of alternate digits in a number.

I believe the concept behind the divisibility rule is clear to you now!

Regards,
Intern
Joined: 21 Apr 2013
Posts: 1
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Computer Software)

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20 Jul 2013, 07:19
1
That was an extremely useful one. Thanks !
Manager
Joined: 10 May 2014
Posts: 138

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30 Dec 2014, 10:29
1
The First Rule of Divisibility is "you do no talk about Divisibility"

The Second Rule of Divisibility is "you do not talk about Divisibility"

I´m just a cinephile and I couldn´t help it... Great post by the way, cyberjadugar!
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Joined: 09 Sep 2013
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24 Oct 2018, 18:16
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Divisibility Rules   [#permalink] 24 Oct 2018, 18:16
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