" 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 \(m+1\).

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 \(m\)."

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?

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 \(m\)

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TopNewest" 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 \(m+1\).

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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 \(m\)."

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I got it.Nice explanation;however,i am not getting what do you mean by

the same result mod mandexactly same mod m. We are supposed to saywith respect to A.Do you meanwith respect to Awhile saying it?Log in to reply

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 \(m\)

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