Number Theory

Prime Factorization and Divisors

Prime Factorization and Divisors: Level 3 Challenges

         

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

Find smallest positive aa for a=b2=c3=d5a={ b }^{ 2 }={ c }^{ 3 }={ d }^{ 5 } such that a,b,c,da,b,c,d are distinct integers.

6128704063125000=23×35×57×79 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 aa and bb , the equation 32a+1=b2{ 3\cdot 2 }^{ a }+1={ b }^{ 2 } has solutions (a1,b1),(a2,b2),...,(an,bn)({ a }_{ 1 },{ b }_{ 1 }),({ a }_{ 2 },{ b }_{ 2 }),...,({ a }_{ n },{ b }_{ n }).

Find a1+b1+a2+b2+...+an+bn{ a }_{ 1 }{ +b }_{ 1 }+{ a }_{ 2 }+{ b }_{ 2 }+...+{ a }_{ n }+{ b }_{ n }?

Given that x2+5x+6x^{2} + 5x + 6 is a prime number, determine the smallest integer value of xx.

Details and Assumptions:

  • If you think no such solution exists, input 999999 as your answer.
×

Problem Loading...

Note Loading...

Set Loading...