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

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

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

2. ## 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?

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

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?

4. ## 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 $\sigma_0(n)$. According to Wikipedia, if $n=\prod_{i=1}^r p_i^{a_i}$ for prime $p_i$'s, then $d(n)=\prod_{i=1}^r(a_i+1)$.

5. ## 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...

6. ## 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.

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

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.

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

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