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

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

    Quote Originally Posted by Victoriaaaa View Post
    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?
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    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).
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    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...
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    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.
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    Re: Stanford Summer program's requirement on taking the math course!!! Please do help

    Quote Originally Posted by Victoriaaaa View Post
    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}.
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