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Disquisitiones Arithmeticae


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Note by Anandmay Patel
11 months, 3 weeks ago

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" Any number \(A\) will be \(0\) or \(1\) or \(2\)................or \(m-1\) modulo \(m\); which account to a total of \(m\) different possibilities.

Also; in a series of \(m\) consecutive integers; two of them cannot have the same result modulo \(m\) since that would make the difference between them a multiple of \(m\) and hence total numbers in the series will become at least \(m+1\).

Yatin Khanna - 11 months, 3 weeks ago

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(Continued)

But, since no two share same result modulo \(m\) and there are \(m\) numbers and only \(m\) possibilities hence; each possibility must occur exactly once and hence; all \(A\) will have one and exactly one number in the series which is exactly the same modulo \(m\)."

Yatin Khanna - 11 months, 3 weeks ago

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I got it.Nice explanation;however,i am not getting what do you mean by the same result mod m and exactly same mod m. We are supposed to say with respect to A.Do you mean with respect to A while saying it?

Anandmay Patel - 11 months, 3 weeks ago

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I havent yet started modular arithmetic formally so I dont know the exact terns for that;
Same result modulo \(m\) means leaves the same remainder when divided by \(m\)
Exactly same mod \(m\) means leaves the same remainder as A leaves when divided by \(m\)

Yatin Khanna - 11 months, 3 weeks ago

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@Yatin Khanna No problem.Thanks

Anandmay Patel - 11 months, 3 weeks ago

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