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## Prime Factorization and Divisors

Learn how to break down numbers big and small, as proposed by the Fundamental Theorem of Arithmetic. See more

# Level 3

Find the number of odd divisors of $$7!$$

Find smallest positive $$a$$ for $$a={ b }^{ 2 }={ c }^{ 3 }={ d }^{ 5 }$$ such that $$a,b,c,d$$ are distinct integers.

$\large 6128704063125000=2^3\times 3^5 \times 5^7 \times 7^9$

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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