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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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