Suppose the first row of a $100\times 100$ matrix is:

$$a_1,a_2,\dots,a_{100}$$

...and suppose that subsequent rows are obtained from the first row by shifting the previous row by $k$ positions.

Therefore, the second row is:

$$a_{k+1},a_{k+2},\dots,a_k$$

For what value of $k$ does this construction produce a Latin square?

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