The answer is $N\ne1,2,4$. Call an integer $n$ good if we can tile an $n\times n$ board with tiles of sizes $5\times5$ and $1\times3$. Clearly $1,2,4$ are not good because we cannot use any $5\times5$ tiles and the number of cells is not divisible by 3. Obviously 3 and 5 are good, and 7 is also good (see attachment). If $n$ is good then $n+3$ is good because we can simply add $1\times3$ tiles to cover the excess. Hence $3,6,9,12,\ldots$ are good, $5,8,11,14,\ldots$ are good, $7,10,13,16,\ldots$ are good. Our proof is complete.

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