Sweet Building

Number Theory Level 2

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 $101^\text{th}$ room.

Then the queen ant decided to build bigger cubic blocks of $5\times 5\times 5$ sugar cubes from all they had previously collected. How many sugar cubes would remain after all these build-ups?

The answer is 1.

1 solution

Worranat Pakornrat
May 27, 2016

Relevant wiki: Euler's Theorem

The total amount of sugar cubes = $1 + 2 + 2^2 + 2^3 + ... + 2^{100} = 2^{101} - 1$ .

According to Euler's theorem , $2^{\phi(5^3)} \equiv 1 \pmod {5^3}$ .

And $\phi(5^3) = 5^3 - 5^2 = 100$ .

Hence, $2^{100} \equiv 1 \pmod {5^3}$ .

$2^{101} \equiv 2 \pmod{5^3}$ .

Thus, $2^{101} - 1 \equiv 1 \pmod{5^3}$ .

As a result, there would be $\boxed{1}$ sugar cube left over after all block build-ups.

Moderator note:

Nice usage of the geometric progression along with Euler's Theorem.

Dull would be his soul who can pass by such a magnificent solution.

Sayan Das - 4 years, 9 months ago

Sweet Building

The answer is 1.

1 solution

Moderator note:

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