Fermat Would Approve Mystery Cache

Around 1637, Fermat was working with an old book by Diophantus called Arithmetica, he scribbled the following in the margin

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

This roughly translates to "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

In simpler sense, Fermat said the he had proved that a^n + b^n != c^n for n > 2. Unfortunately, the proof was lost to history. The closest we have from Fermat is the proof for n=4.

After Fermat's death in 1665, his margin notes were printed in a new edition of Arithmetica, starting off what would be a more than 300 year journey to the proof. Many mathematicians worked to provide a generalized proof including Sophie Germain, Ken Ribet, Ernst Kummer, and Gerhard Frey. In the end, it wasn't until 1993 that a generalized proof was published by Andrew Wiles (this work was later confirmed by two papers he published in 1994 and 1995.)

Some argue that Fermat could not have made such a proof as Wiles created without the use of advanced computers... "a 20th century proof" it was called.

In the end, I think Fermat would approve of this cache as it doesn't break his rules. As usual, you're looking for only the last digit in each set.

You can validate your puzzle solution with certitude.