
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
What is the remainder when \(2^{2015}\) is divided by 20? Considering it is difficult to perform this division directly using only paper and pencil, we will search for a way to obtain the answer by analyzing the remainders of powers of 2 modulo 20, hoping to find a pattern.

What is the remainder when \(2^{5}\) is divided by 20?

\[2^{5} = 32 = 1 \cdot 20 + 12\]

Knowing this, it's easy to determine the remainder of \(2^{6}\) modulo 20, as

\[2^{6} = 2 \cdot 2^{5} = 2 \cdot (1 \cdot 20 + 12) = 2 \cdot 20 + 24\]

Since 24 is greater than 20 and cannot be a remainder, we must write

\[2^{6} = 3 \cdot 20 + 4\]

We can extend the previous argument concluding that to determine the remainder of \(2^{i+1}\) modulo 20, it is enough to know the remainder of the product of the remainder of \(2^{i}\) modulo 20. This way, we can build the sequence of powers and remainders modulo 20.

\[
\begin{array}{ccccccc}
n & 2^{1} & 2^{2} & 2^{3} & 2^{4} & 2^{5} & 2^{6} \\
\text{Remainder modulo 20} & 2 & 4 & 8 & 16 & 12 & 4
\end{array}
\]

(a) Determine the remainders when \(2^{7}, 2^{10},\) and \( 2^{13} \) are divided by 20.

(b) Knowing that the remainders repeat periodically, determine the period of repetition, that is, the number of distinct remainders that keep repeating.

(c) Return to the initial question. What is the remainder when \(2^{2015}\) is divided by 20?

Present the answer in LaTex format: \boxed{Your answer}
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