The discussion focuses on calculating the number of divisors of the expression $2^2 \cdot 3^3 \cdot 5^3 \cdot 7^5$ that are of the form $4n+1$ and the total number of positive divisors of $7!$ that are of the form $3t+1$. The divisor count for $2^2 \cdot 3^3 \cdot 5^3 \cdot 7^5$ can be determined using properties of quadratic residues, while the divisors of $7!$ can be analyzed through its prime factorization and congruences. Both calculations require a solid understanding of number theory and divisor functions.
PREREQUISITESMathematicians, students of number theory, and anyone interested in advanced divisor counting techniques and modular arithmetic applications.
jacks said:(1) The number of divisers of the form $2^2.3^3.5^3.7^5$ which are is in the form of $4n+1$ where $n\in\mathbb{N}$
(2) Calculate Total no. of positive Divisers of $7!$ which are is in the form of $3t+1\;,$ where $t\in \mathbb{N}$