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A question defines A = {1,2,3,4,5,6} and a binary operation * such that $a*b=r$, where r is the least non-negative remainder when the product $ab$ is divided by $k$. Find k for * to be a binary operation.

The answer is five, apparently, but;

1: What's up with the "non-negative remainder" thing? Are remainders ever negative?

2: On trying out 1 and 5 as $a$ and $b$, the remainder on dividing it by k(which is 5) is 0. And if 0 is acceptable as th least possible remainder, then k wouldn't be 5, it would be 1, and the remainder in each case would be zero, wouldn't it?

Why did it work out this way?

harry
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  • From my understanding $k$ could be anything---there's no constraint on $k$---so I'm confused. –  Oct 29 '20 at 06:40
  • The question phrases it as a definite constant answer, but everything else in it doesn't help at all. – harry Oct 29 '20 at 06:44
  • @Harry Holmes The question is phrased in an appallingly poor fashion (and the fact that this approach tends to be rather mainstream-ish in some textbooks is even more appalling). The goal of the question is to find the $k$ for which the (a priori informally defined) correspondence described in the text would eventually allow one to rigorously define a binary operation on $A$. – ΑΘΩ Oct 29 '20 at 06:50
  • Sheesh. There isn't an answer or explanation, then? – harry Oct 29 '20 at 06:58
  • @Harry Holmes Allow me to add a few more observations: 1) you are most right in noting that remainders upon division by $k \in \mathbb{N}^{\times}$ are necessarily in the natural interval $[0, k-1]$ 2) you are also correct in your reasoning concerning the situation $k=5$ 3) you would be successful in defining a binary operation on $A$ along the specified procedure when considering $k=7$; this itself is a rather profound -- albeit elementary -- fact of abstract algebra, and it relates directly to the multiplicative group $\mathbb{Z}_7^{\times}$ of the field $\mathbb{Z}_7$. – ΑΘΩ Oct 29 '20 at 12:34
  • @Harry Holmes And -- if I may -- one final word by which I mean to encourage you: waste not your time with mathematical texts of dubious quality and focus instead on treatises/textbooks which display a solid spirit of rigour. Judging by what you describe regarding the question, its source does not appear to belong to this latter category of texts (if it originates not from a book but from a person, there is definite room for improvement in how the respective person frames a problem). – ΑΘΩ Oct 29 '20 at 12:37

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