Inference of even or odd from the number of divisors?

[MathematicalPhysicist]

is there some theorem of some sort, that connect the number of divisors of a number to identify if it's even or odd?

or to be more specific, does number of divisors of a number has any significance in number theory?

 

[shmoe]

is there some theorem of some sort, that connect the number of divisors of a number to identify if it's even or odd?

No. Number of divisors depends only on the exponents in the prime factorization of a number, it doesn't care what these primes are.

Determining if a number is even is not generally a difficult thing to do at any rate.

ior to be more specific, does number of divisors of a number has any significance in number theory?

Yes, the number of divisors function and many variants are studied extensively.

 

[MathematicalPhysicist]

Yes, the number of divisors function and many variants are studied extensively.

what examples of number of divisors function can you give?

 

[shmoe]

The basic one counts the number of ways to write n as a product of 2 numbers, you can consider number of ways to write it as a product of k numbers. You can also consider the sum of the divisors, sum of squares of the divisors, etc.

 

[Gokul43201]

lqg, look up the following :

number theoretic functions, the tau function, the divisor function - Mathworld is one place to start

Note : The number of divisors can tell you whether or not a number is a perfect square