Question

the integer N is positive. There are exactly 2005 ordered pairs (x,y) of positive integers satisfying: 1/x+1/y=1/N Prove that N is a perfect square

Answer 1

@anhkhoavo1210
i dont wanna give a complete solution...follow the hints

1.given equation is equaivalent to \((x-N)(y-N)=N^2 \ \ (\star)\) (why?)

2. \(x>N\) and \(y>N\) ...(why?) so \(x-N\) and \(y-N\) are both positive integers

3. show that number of solutions for equation \((\star)\) is equivalent to number of divisors of \(N^2\)

4. let \( N= p_1^{\alpha_1} \times p_2^{\alpha_2}\times...\times p_k^{\alpha_k} \) and show that number of divisors of \(N^2=\prod_{i=1}^{k} (2\alpha_i+1)\)

5.so we have \(\prod_{i=1}^{k} (2\alpha_i+1)=2005=5 \times 401\)

6. from 6 show that \(k=1 \ and \ \alpha_1=1002\) or \(k=2 , \alpha_1=2 \ and \ \alpha_1=200\)

Answer 2

* 6. from 5 show....

Answer 3

Very nice. thanks