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Euler's Totient Function, often denoted as \( \phi(n) \), is an important concept in number theory. It helps determine the number of integers up to a given integer \( n \) that are coprime to \( n \).
For a number \( n \) that can be expressed as a product of two prime numbers \( p \) and \( q \), as in the given problem, the totient function formula is \( \phi(n) = (p-1)(q-1) \).
This function is crucial for problems involving factorization of numbers using prime numbers, like the given exercise.
It lays the groundwork for cryptosystems like RSA, where knowing the totient is important for encryption and decryption processes. To utilize \( \phi(n) \) efficiently, it's key to recognize the properties of prime factors:

• For any prime \( p \), \( \phi(p) = p-1 \).
• The product formula for \( \phi(n) = \phi(p) \cdot \phi(q) \) holds if \( n = p \cdot q \).

Quadratic equations appear frequently in solving problems related to number theory, like the current one where the equations \( p+q \) and \( (p+q)^2-4n \) are utilized. A quadratic equation is typically expressed in the form \( ax^2 + bx + c = 0 \).
This exercise indirectly requires solving a system akin to quadratic equations.Given:

• \( p+q \) simplifies to one equation based on the characteristics of \( n \) and \( \phi(n) \).
• \( p-q = \sqrt{(p+q)^2 - 4n} \), another equation, resembles working out quadratic solutions.

Combining these equations can help find solutions for \( p \) and \( q \). The methodology follows the techniques used in solving quadratic systems with variables transformed into simpler expressions for easier interpretation.

Understanding perfect squares helps immensely when solving the quadratic derivation in the exercise. A perfect square is a number that is the square of an integer. In the problem, the difference calculated \( \sqrt{2528100} = 1590 \) is a perfect square, greatly simplifying the problem.Here's why recognizing perfect squares matters:

• Direct calculation of square roots is simpler when dealing with perfect squares.
• Perfect squares reveal hidden integer solutions in equations, helping solve for variables like \( p \) and \( q \).
• Solving \( p-q \) uses the square root of \( 2528100 \), showing it is crucial to understanding its perfect square nature.

Being attentive to perfect squares and their properties allows you to simplify and solve equations that might otherwise seem more complex.