Скажи мне, где ты был прошлой ночью.

Can this also be “прошлою” or is that not valid?

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Google Ngram Viewer for Russian shows that the frequency of прошлою is very low, far lower than that of прошлой:

English Translation

Tell me where you were last night.

I believe “прошлою” could also be used, although it is nowadays considered archaic/poetic. Feel free to correct me, native Russian speakers.