Number Theory
# Prime Factorization and Divisors

Find the number of odd (positive) divisors of \(7!\)

\[ 6128704063125000=2^3\times 3^5 \times 5^7 \times 7^9\]

For the number above, find the number of divisors which are perfect squares.

For two non-negative integers \(a\) and \(b\) , the equation \({ 3\cdot 2 }^{ a }+1={ b }^{ 2 }\) has solutions \(({ a }_{ 1 },{ b }_{ 1 }),({ a }_{ 2 },{ b }_{ 2 }),...,({ a }_{ n },{ b }_{ n })\).

Find \({ a }_{ 1 }{ +b }_{ 1 }+{ a }_{ 2 }+{ b }_{ 2 }+...+{ a }_{ n }+{ b }_{ n }\)?

Given that \(x^{2} + 5x + 6\) is a prime number, determine the smallest integer value of \(x\).

**Details and Assumptions**:

- If you think no such solution exists, input \(999\) as your answer.

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