Stanford Summer program's requirement on taking the math course!!! Please do help!

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- How many pairs of positive integers
*x *and *y *satisfy the equation *x**y **= *20132013 ? How many pairs ofpositive integers *x *and *y *satisfy the equation *x**y **= *20122012?

Re: Stanford Summer program's requirement on taking the math course!!! Please do help

How many pairs of positive integers x and y satisfy the equation xy = 20132013 ? How many pairs of positive integers x and y satisfy the equation xy = 20122012?

Re: Stanford Summer program's requirement on taking the math course!!! Please do help

Quote:

Originally Posted by

**Victoriaaaa** How many pairs of positive integers x and y satisfy the equation xy = 20122012?

Look at this webpage.

It tells you that there are 24 divisors. So what is the answer?

Re: Stanford Summer program's requirement on taking the math course!!! Please do help

If you are talking about ordered pairs (x, y) such that xy = n, then their number equals the number of divisors of n, which is denoted by d(n) or $\displaystyle \sigma_0(n)$. According to Wikipedia, if $\displaystyle n=\prod_{i=1}^r p_i^{a_i}$ for prime $\displaystyle p_i$'s, then $\displaystyle d(n)=\prod_{i=1}^r(a_i+1)$.

Re: Stanford Summer program's requirement on taking the math course!!! Please do help

How many pairs of positive integers x and y satisfy the equation x^y = 2013^2013 ? How many pairs of positive integers x and y satisfy the equation x^y = 2012^2012?

Sorry, I think I didn't make the question clear...

Re: Stanford Summer program's requirement on taking the math course!!! Please do help

How many pairs of positive integers x and y satisfy the equation x^y = 2013^2013 ? How many pairs of positive integers x and y satisfy the equation x^y = 2012^2012?

Sorry, I think I didn't make the question clear... It not just divisor.

Re: Stanford Summer program's requirement on taking the math course!!! Please do help

Quote:

Originally Posted by

**Victoriaaaa** How many pairs of positive integers x and y satisfy the equation x^y = 2013^2013 ? How many pairs of positive integers x and y satisfy the equation x^y = 2012^2012?

Sorry, I think I didn't make the question clear... It not just divisor.

Well shame on you.

$\displaystyle 2012=2^2\cdot 503$ so that $\displaystyle 2012^{2012}=2^{4024}\cdot 503^{2012}$

There are $\displaystyle (4024+1)(2012+1)$ actual divisors of $\displaystyle 2012^{2012}$.