DIVISIBILITY TESTS

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1 Aug 2008 09:40:39 IST

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1 Aug 2008 09:40:39 IST

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DIVISIBILITY TESTS

Mathematics , Algebra

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 10equiv0 mod 2
And
So xequiv 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.
10equiv1 mod 3
And x=a_0+10a_1+10^2a_2....................10^ma_m
So xequiv a_0+1a_1+1^2a_2....................1^ma_m mod 3
Rightarrow xequiv 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.
100equiv0 mod 4
And x=a_0+10a_1+10^2a_2....................10^ma_m
So xequiv a_0+10a_1+0a_2....................0a_m mod 4
Rightarrow xequiv 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 10equiv0 mod 5
And x=a_0+10a_1+10^2a_2....................10^ma_m
So xequiv a_0+0a_1+0^2a_2....................0^ma_m mod 5
Rightarrow xequiv 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.
10equiv1 mod 9
And x=a_0+10a_1+10^2a_2....................10^ma_m
So xequiv a_0+1a_1+1^2a_2....................1^ma_m mod 9
Rightarrow xequiv 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
10equiv0 mod 10
And
So xequiv a_0+0a_1+0^2a_2....................0^ma_m mod 10
Rightarrow xequiv a_0 mod 10
Hence a number is divisible by 10 if the last digit is divisible by 10.

DIVISIBILITY TEST FOR 11
10equiv-1 mod 11
And x=a_0+10a_1+10^2a_2....................10^ma_m
So xequiv a_0+(-1)a_1+(-1)^2a_2....................(-1)^ma_m mod 11
Rightarrow xequiv a_0-a_1+a_2....................(-1)^ma_mmod 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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Comments (10)

sahil madaan

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1 Aug 2008 10:19:17 IST

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good one!!

Nikhil Bagade

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1 Aug 2008 11:13:26 IST

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nice rey!!!!!!

Protyush Sahu

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1 Aug 2008 13:28:27 IST

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wonderful

Naruto Uzumaki

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1 Aug 2008 22:46:47 IST

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marvellous

jithmeister

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1 Aug 2008 23:16:42 IST

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Nice. Food for thought.

krish1092

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2 Aug 2008 18:25:44 IST

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nice work

bharat ravi

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2 Aug 2008 19:46:27 IST

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gud

®µD®A

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3 Aug 2008 10:51:54 IST

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thank you.

Anay Pattanaik

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4 Aug 2008 14:17:13 IST

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very gud

Shreyaarya

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27 Aug 2008 20:31:19 IST

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gr88...gud work

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