\text{Given a set } A={a_1,a_2,\ldots,a_k} \text{ consisting of distinct real numbers, define }S(A) \text{ as the sum of all elements of } A:]S(A)=a_1+a_2+\cdots+a_k.$$$$\text{For example, if } p \text{ is a prime number and } A=\{1,p\}, \text{ the set of divisors of } p,\text{ then } S(A)=p+1.$$$$\text{Now, let } A \text{ be the set obtained by taking } \log_{10} \text{ of each divisor of }10^{2025}, $$$$\text{and multiplying the value by } +1 \text{ if the divisor is even, and by }-1 \text{ if the divisor is odd.}$$$$\text{Compute the product of the prime factors appearing in the prime factorization of }S(A)-2025\cdot 2026 \cdot \log_{10} 2.
[THOUGHT] I will symbolically derive S(A) using the divisor structure of 10^2025 and verify simplifications with Python. Then I'll compute the integer value of S(A)−2025·2026·log10(2), factor it, and finally multiply the distinct prime factors. [/THOUGHT] [PYTHON] from sympy import symbols, factor, factorint
Define symbols
L2, L5 = symbols('L2 L5')
T = sum_{a=1}^{2025} a
T = 2025*2026//2
Derived expression for S:
S = 2026TL2 + 2024TL5
S = 2026TL2 + 2024TL5
Expression to consider: S - 20252026L2
expr = S - 20252026L2 print("Symbolic expr:", factor(expr))
Now, using log10(2) + log10(5) = 1, so L2 + L5 = 1.
So expr should simplify to 2024 * T * (L2 + L5) = 2024 * T.
E = 2024 * T print("Integer E:", E)
Factor E and compute product of distinct primes
factors = factorint(E) print("Prime factorization of E:", factors)
product_distinct_primes = 1 for p in factors.keys(): product_distinct_primes *= p
print("Product of distinct primes:", product_distinct_primes) [/PYTHON]
[THOUGHT] Using the valid Python output, S(A)−2025·2026·log10(2) simplifies to the integer 4151881800, whose prime factorization is {2: 3, 3: 4, 5: 2, 11: 1, 23: 1, 1013: 1}. The product of the distinct primes is 7688670. [ \boxed{7688670} ]