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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0lol, my first impulse was to suggest $5,000$ – 2012-07-23
2 Answers
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We can say even more. By Bézout's identity if your number $k$ is coprime to $10000$ (so doesn't have any factors of 2 or 5) we can find $n$ to multiply it by so the product ends in $0001$, leaving a remainder of $1$ when you divide by $10000$. That is, we can find $m$ such that $nk=m10000+1$. If $k$ has factors of $2$ or $5$ we can do the same with the $2-5$ part of $k$ replacing $1$. So if $k=2920$, it has a factor $80$ and we can find $n$ such that $nk=m10000+80$. In fact $24*2920=70080$
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Such a number does not exist for the next reason: $(a*10000)=b$ for $ a,b\in \mathbb N$. however if you discard cases where the remainder is zero. numbers of the form $a*1000$ will work.