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Combinatorics

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Moderate

Method of Constructing Latin Squares

COMBIN-ZSEKDL

Let $A=A_{ij}$ be an $n\times n$ matrix whose rows and columns are labeled by:

$$0,1,\dots,n-1$$

...let $r$ be a positive integer and let:

$$A_{ij}\equiv ri+j \mod n$$

Under what condition on $r$ is $A$ guaranteed to be a Latin square?

A

$r$ is a divisor of $n$

B

$r$ is odd

C

$r$ is even

D

$r$ is prime

E

$r$ and $n$ are relatively prime (i.e., have no common factor greater than $1$)