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March 29, 2024, 01:56:00 am

Author Topic: 216 bus  (Read 1363 times)  Share 

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kamil9876

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216 bus
« on: June 13, 2009, 08:02:28 pm »
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Prove that if is greater than and divisible by , but not divisible by , then:

is the remainder when is divided by where is any natural number.
« Last Edit: June 13, 2009, 11:39:33 pm by kamil9876 »
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

Mao

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Re: 216 bus
« Reply #1 on: June 14, 2009, 12:10:16 pm »
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Let , where m is an integer greater than 1





Eventually, this gives



The remainder is hence

QED.
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kamil9876

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Re: 216 bus
« Reply #2 on: June 14, 2009, 12:57:25 pm »
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This was inspired by the 216 bus. Seeing it reminded me that but so i wondered if all ends in a 6. I proved it to be true and my original post is the generalisation for any base . 10 is a possible base because it is only divisible by 2 once. I think my proof was a proof by induction, but i cannot remember as I haven't put it down on paper yet:P didn't have any with me when i was on the 216 bus
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."

kamil9876

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Re: 216 bus
« Reply #3 on: June 14, 2009, 09:56:50 pm »
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ok so this is what I came up with:

is even.

assume that it's true for k=a, we wish to prove it is true for a+1:






is a multiple of because is even.

That means our number can be written as:



As required.

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This was inspired from the proof that always ends in for natural in base 10. (the case for n=10):



First term is an even multiple of half of ten, meaning its a multiple of 10, hence the addition of 6 makes it the last digit. Generalizing this:

if



The first and third term obviously end in 0. The middle term is an even multiple of half of ten, meaning that it's a multiple of 10 so it too ends in 0, hence the addition of 6 makes the last digit 6.

This is analogous to the more generalized version posted originally, which is just an extension of this proof to bases other than 10.(that are only divisible by 2 once)
Voltaire: "There is an astonishing imagination even in the science of mathematics ... We repeat, there is far more imagination in the head of Archimedes than in that of Homer."