I am looking for a number that when multiplied by any number and divided by 10000 never leaves the 3 digit number as 291 , I mean I am looking for a number that leaves a remainder as 1231 , 1001 ...etc and not like 0012,0291,0001.
Unique number for multiplication
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elementary-number-theory
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0I am not sure about what you are asking. Are you looking for a number $N$ such that the division algorithm gives $Nk=10000q+r$ with $1000\leq r\leq9999$ for all $k>0$ ? As stated, this is impossible: for any $N$ take $k=10000$. – 2012-07-23
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0I am asking a number say "n" which when multiplied by a number say 1221 and divided by 10000 the remainder must be a four digit remainder like 1234 and not like 0001 (which is a single digit number) – 2012-07-23
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0As I said, this is impossible unless you give some restrictions on the "multiplier" (like $1221$ in your comment) – 2012-07-23
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2You can break this into two cases: 1) $gcd(n, 10,000) = 1$ and 2) $gcd(n, 10,000) \neq 1$. In the second case, you can clearly see that for some $k,\ kn = 0 (\text{mod}\ 10,000)$. In the first case, there will be some $k$ with $kn = 1 (\text{mod}\ 10,000)$. So, no such $n$. – 2012-07-23
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1So you won't accept remainder 0 either? Otherwise numbers like $1000$, $1250$, $2500$, $5000$, or $10000$ itself would work. But it is clearly impossible to not get zero. Multiply by $10000$ :-) – 2012-07-23
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0lol, my first impulse was to suggest $5,000$ – 2012-07-23