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Topic: Can you prove this ?
Replies: 4   Last Post: Jun 30, 1996 7:25 PM

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Dan Wevrick

Posts: 55
Registered: 12/6/04
Re: Can you prove this ?
Posted: Jun 26, 1996 10:41 PM
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In article <d93-hfa.835814513@nada.kth.se>,
HÃ¥kan Fahlstedt <d93-hfa@nada.kth.se> wrote:
>Many years ago a math teacer told me that if you add all the digits in a
>number and the sum is divisible by 3 then the number is also divisible by 3.
>
>Can you prove this in any way and if so can you find more rules like this with
>other numbers.


Well, 10^k == 1 (mod 3) for k=1,2,,,

If n = a_0 + a_1*10 + a_2*10^2 + ... a_k*10^k, where a_0, a_1,..., a_k are
in {0,1...9} (ie. the units digit of n is a_0, the tens digit of n is a_1 etc)

then n == 0 (mod 3) (ie. n is divisible by 3)
iff a_0 + a_1*10 + a_2*10^2 + ... + a_k*10^k == 0 (mod 3)
iff a_0 + a_1 + a_2 + ... + a_k == 0 (mod 3)

Some other results.

n is divisible by 2 : if a_0 is in {0,2,4,6,8}
n is divisible by 3 : if the sum of the digits is divisible by 3
n is divisible by 4 : if the last two digits are divisible by 4
n is divisible by 5 : if a_0 is 0 or 5

There are a few more.

Dan







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