An army of worker ants was carrying sugar cubes back into their colony. In there, the ants put 1 sugar cube into the first room, 2 into the second, 4 into the third, and doubling the amount so on until the room.
Then the queen ant decided to build bigger cubic blocks of
sugar cubes from all they had previously collected. How many sugar cubes would remain after all these build-ups?
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Relevant wiki: Euler's Theorem
The total amount of sugar cubes = 1 + 2 + 2 2 + 2 3 + . . . + 2 1 0 0 = 2 1 0 1 − 1 .
According to Euler's theorem , 2 ϕ ( 5 3 ) ≡ 1 ( m o d 5 3 ) .
And ϕ ( 5 3 ) = 5 3 − 5 2 = 1 0 0 .
Hence, 2 1 0 0 ≡ 1 ( m o d 5 3 ) .
2 1 0 1 ≡ 2 ( m o d 5 3 ) .
Thus, 2 1 0 1 − 1 ≡ 1 ( m o d 5 3 ) .
As a result, there would be 1 sugar cube left over after all block build-ups.