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Euler's Totient Function, denoted as \(\varphi(n)\), is a key concept in number theory that counts the number of positive integers up to a given integer \(n\) that are coprime to \(n\). Two numbers are coprime if their greatest common divisor is 1. This function is particularly useful in various fields, including cryptography and RSA encryption.

If \(n\) is a prime number, then \(\varphi(n)\) is simply \(n - 1\) because all positive integers less than \(n\) are coprime to \(n\). For example, \(\varphi(7) = 6\), since all numbers from 1 to 6 are coprime with 7.

When \(n\) is not a prime, Euler's Totient Function becomes more complex. Fortunately, due to its multiplicativity, \(\varphi(n)\) for a composite number is calculated by multiplying the totients of its prime factors. Specifically, if \(n = p_1^{e_1} p_2^{e_2} \ldots p_r^{e_r}\) where \(p_i\) are distinct primes, the formula becomes:
\[\varphi(n) = \varphi(p_1^{e_1}) \varphi(p_2^{e_2}) \cdots \varphi(p_r^{e_r})\]

Each prime power is evaluated individually as \(\varphi(p^k) = p^k - p^{k-1}\), which simplifies to \(p^{k-1}(p-1)\). Understanding this function aids in solving problems related to divisibility and number theory property exploration.

Prime factorization is the process of expressing a number as the product of its prime factors. It's a fundamental concept of number theory and indispensable for understanding properties such as divisibility and working with Euler's Totient Function.

For instance, given a number \(n\) divisible by distinct odd primes, we express \(n\) as \(n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r}\). Here, \(p_i\) represent the distinct prime numbers and \(e_i\) count their occurrences.

Each unique scenario calls for distinct use of their prime powers, often seen in problems involving totients. As in the given exercise, acknowledging \(n\)'s prime factorization allows us to break \(\varphi(n)\) down into manageable parts. This aids in more complex proof scenarios, like demonstrating divisibility in terms of powers of 2 or other numbers.

Knowing this breakdown becomes crucial when combining understanding with Euler's Totient Function, as seen where each distinct prime's power influences the calculation of \(\varphi(n)\). The multiplicative nature of the totient function means having clarity on prime components simplifies complex number theory operations immensely.

Divisibility is a key concept in understanding how one integer divides another without leaving a remainder. In number theory, divisibility helps explore properties such as evenness, factors, and closely involves concepts like Euler's Totient Function.

From the exercise, we see that if \(n\) is divisible by \(r\) distinct odd primes, it has a profound effect on \(\varphi(n)\) such that \(2^r\) divides \(\varphi(n)\). This stems from knowing that subtracting 1 from any odd prime \(p_i\) results in an even number. Consequently, each prime factor contributes a factor of 2 to \(\varphi(n)\), which collectively results in \(2^r\) in the product representation.

This visualization of divisibility helps confirm that \(2^r\) exactly divides \(\varphi(n)\), without oversight. Divisibility becomes more intuitive with patterns distinguished in the multiplication and totient application, highlighting its importance in proofs and mathematical logic.