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DIVISIBILITY TESTS

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Tagged with: academic

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posted on 1 Aug 2008 09:40:39 IST

Friends I will prove all the divisibility.

I know you know all of them but just for those who don't know can see this.

Let the number be $x=a_ma_{m-1}a_{m-2}..............a_2a_1a_0$

As a₀ is the unit digit of the number then the number can be written as x=a_0+10a_1+10^2a_2....................10^ma_m

DIVISIBILITY TEST FOR 2.

A number is divisible by 2 if the last digit is divisible by 2.

We know $10\equiv0\ mod\ 2$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+0a_1+0^2a_2....................0^ma_m\ mod\ 2$

$\Rightarrow x \equiv a_0\ mod\ 2$

Hence a number is divisible by 2 if the last digit is divisible by 2.

DIVISIBILITY TEST FOR 3.

A number is divisible by 3 if the sum of the digits is divisible by 3.

$10\equiv1\ mod\ 3$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+1a_1+1^2a_2....................1^ma_m\ mod 3$

$\Rightarrow x\equiv a_0+a_1+a_2....................a_m\ mod\ 3$

Hence A number is divisible by 3 if the sum of the digits is divisible by 3.

DIVISIBILITY TEST FOR 4

A number is divisible by 4 if the last two digits are divisible by 4.

$100\equiv0\ mod\ 4$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+10a_1+0a_2....................0a_m\ mod\ 4$

$\Rightarrow x\equiv a_0+10a_1\ mod\ 4$

Hence A number is divisible by 4 if the last two digits are divisible by 4.

DIVISIBILITY TEST FOR 5.

A number is divisible by 5 if the last digit is divisible by 5.

We know that $10\equiv0\ mod\ 5$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+0a_1+0^2a_2....................0^ma_m\ mod\ 5$

$\Rightarrow x\equiv a_0\ mod\ 5$

Hence a number is divisible by 5 if the last digit is divisible by 5.

DIVISIBILITY TEST FOR 9.

A number is divisible by 9 if the sum of the digits is divisible by 9.

$10\equiv1\ mod\ 9$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+1a_1+1^2a_2....................1^ma_m\ mod\ 9$

$\Rightarrow x\equiv a_0+a_1+a_2....................a_m\ mod\ 9$

Hence A number is divisible by 9 if the sum of the digits is divisible by 9.

DIVISIBILITY TEST FOR 10

$10\equiv0\ mod\ 10$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+0a_1+0^2a_2....................0^ma_m\ mod\ 10$

$\Rightarrow x\equiv a_0\ mod\ 10$

Hence a number is divisible by 10 if the last digit is divisible by 10.

DIVISIBILITY TEST FOR 11

$10\equiv-1\ mod\ 11$

And x=a_0+10a_1+10^2a_2....................10^ma_m

So $x\equiv a_0+(-1)a_1+(-1)^2a_2....................(-1)^ma_m\ mod\ 11$

$\Rightarrow x\equiv a_0-a_1+a_2....................(-1)^ma_m\mod\ 11$

I took a lot of time typing it and in the TEXER.

If it have helped then please leave a commentand rate if you liked it.

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rudra.panda (2802)

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Olaaa!! Perrrfect answer.

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sahilmadaan12 is offline

comment by sahilmadaan12 (posted on 1 Aug 2008 10:19:17 IST)

good one!!

GoNik is offline

comment by GoNik (posted on 1 Aug 2008 11:13:26 IST)

nice rey!!!!!!

Protyush_Sahu is offline

comment by Protyush_Sahu (posted on 1 Aug 2008 13:28:27 IST)

wonderful

vibhu.oct is offline

comment by vibhu.oct (posted on 1 Aug 2008 22:46:47 IST)

marvellous

knowmonger is offline

comment by knowmonger (posted on 1 Aug 2008 23:16:42 IST)

Nice. Food for thought.

krish1092 is offline

comment by krish1092 (posted on 2 Aug 2008 18:25:44 IST)

nice work

smilingbharat is offline

comment by smilingbharat (posted on 2 Aug 2008 19:46:27 IST)

gud

rudra.panda is offline

comment by rudra.panda (posted on 3 Aug 2008 10:51:54 IST)

thank you.

anay_pat is offline

comment by anay_pat (posted on 4 Aug 2008 14:17:13 IST)

very gud

shreyaarya is offline

comment by shreyaarya (posted on 27 Aug 2008 20:31:19 IST)

gr88...gud work

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