by Don  

The word прошлый means 'last' in the sense of «В прошлом году я перестал чистить зубы» "Last year I stopped brushing my teeth" or «В прошлом месяце у меня выпали все зубы» "Last month all my teeth fell out." Прошлый means last in the sense of time but not in the sense of "last one of the batch." In other words, you can use it to say 'last week' but you can't use it to say 'I bought the last tomato.'

We particularly find this word in the following time phrases:

last year в прошлом году
last summer прошлым летом
last month в прошлом месяце
last week на прошлой неделе
last weekend в прошлые выходные
last Saturday в прошлую субботу

Notice that years and months are used with the preposition в and the prepositional case (году is an irregular prepositional form), seasons are used in the instrumental case with no preposition, weeks are used with на and the prepositional case, and days are used with в and the accusative case.

On occasion people will say «прошлой субботой» in the instrumental case with no preposition to mean 'last Saturday,' but that's considered uneducated. Never write it in a composition.

Last but not least, if you are discussing American academic life, you will need the phrases:

last semester в прошлом семестре
last quarter в прошлой четверти

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