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Euler's Totient Function, often denoted as \( \phi(n) \), is a fundamental concept in number theory. It measures the count of positive integers up to a given integer \( n \) that are coprime to \( n \). Two numbers are considered coprime if their greatest common divisor (gcd) is 1. For example, \( \phi(9) = 6 \) because precisely six numbers (1, 2, 4, 5, 7, and 8) are coprime to 9.
To calculate \( \phi(n) \), especially for a product of distinct prime factors, we use the formula:

• If \( n = p_1^{k_1} \cdot p_2^{k_2} \cdot \ldots \cdot p_m^{k_m} \), then \( \phi(n) = n \left(1 - \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)\ldots\left(1 - \frac{1}{p_m}\right) \).

Euler's Totient Function has a deep connection to other aspects of number theory, such as modular arithmetic and the RSA encryption algorithm. It also features prominently in this exercise as we leverage its properties to transform sums over divisors.

The Möbius Inversion Formula is a powerful tool in number theory for addressing multiplicative functions. It is tied to the Möbius function \( \mu(n) \), which is defined as follows:

• \( \mu(n) = 1 \) if \( n = 1 \).
• \( \mu(n) = 0 \) if \( n \) has a squared prime factor.
• \( \mu(n) = (-1)^k \) if \( n \) is a product of \( k \) distinct primes.

The Möbius Inversion Formula allows us to express a summatory function based on divisors in terms of the Möbius function:

• If \( f(n) = \sum_{d \mid n} g(d) \), then \( g(n) = \sum_{d \mid n} \mu(d) f\left(\frac{n}{d}\right) \).

In this exercise, the Möbius Inversion Formula helps translate between different sum expressions, shedding light on how the sums interact to produce the observed results, particularly with adjustments for parity and signs corresponding to Euler's Totient values.

Divisors are numbers that divide another number without leaving a remainder. For any integer \( n \), the divisors are the integers \( d \) such that \( n \equiv 0 \mod d \). Understanding divisors is crucial as they play a vital role in various number theoretic functions and summations:

• The sum and product of divisors can often be used to analyze and simplify complex mathematical identities.
• The properties of divisors are used extensively in the hints provided in exercises like this to decompose sums.
• Knowing that every number \( n \) can be expressed in terms of its divisors allows for strategic approaches to solving problems related to number divisibility and congruence.

In the solution specified, exploring sums over divisors helps in understanding how both evenness and oddness (parity) influence the outcomes of the expression sums.

Parity refers to whether an integer is even or odd. An integer \( n \) is even if it can be expressed as \( n = 2k \) for some integer \( k \), and odd if \( n = 2k + 1 \). Parity influences many aspects of number theory as it does the theorem in this exercise.

• When \( n \) is even, it's hinted that certain patterns or symmetries occur in the divisors that tend to cancel out, leading to a sum of 0.
• When \( n \) is odd, the symmetry is disrupted, and the sum does not cancel, resulting in \(-n\).
• Parity frequently appears in the context of functions and sequences due to their inherent property of dividing numbers which directly influences divisibility rules and formulas.

In the problem provided, analyzing parity helps in determining the outcome and proves crucial to understanding the behavior of divisors in the sum given different values of \( n \), both even and odd.