Problem 157
Solving the diophantine equation 1/a+1/b= p/10n
Consider the diophantine equation 1/a+1/b= p/10n with a, b, p, n positive integers and a ≤ b.
For n=1 this equation has 20 solutions that are listed below:
| 1/1+1/1=20/10 | 1/1+1/2=15/10 | 1/1+1/5=12/10 | 1/1+1/10=11/10 | 1/2+1/2=10/10 |
| 1/2+1/5=7/10 | 1/2+1/10=6/10 | 1/3+1/6=5/10 | 1/3+1/15=4/10 | 1/4+1/4=5/10 |
| 1/4+1/20=3/10 | 1/5+1/5=4/10 | 1/5+1/10=3/10 | 1/6+1/30=2/10 | 1/10+1/10=2/10 |
| 1/11+1/110=1/10 | 1/12+1/60=1/10 | 1/14+1/35=1/10 | 1/15+1/30=1/10 | 1/20+1/20=1/10 |
How many solutions has this equation for 1 ≤ n ≤ 9?
解不定方程1/a+1/b= p/10n
考虑不定方程1/a+1/b= p/10n，其中a, b, p, n均为正整数，且a ≤ b。
对于n=1，这个方程有20个解，如下所示：
| 1/1+1/1=20/10 | 1/1+1/2=15/10 | 1/1+1/5=12/10 | 1/1+1/10=11/10 | 1/2+1/2=10/10 |
| 1/2+1/5=7/10 | 1/2+1/10=6/10 | 1/3+1/6=5/10 | 1/3+1/15=4/10 | 1/4+1/4=5/10 |
| 1/4+1/20=3/10 | 1/5+1/5=4/10 | 1/5+1/10=3/10 | 1/6+1/30=2/10 | 1/10+1/10=2/10 |
| 1/11+1/110=1/10 | 1/12+1/60=1/10 | 1/14+1/35=1/10 | 1/15+1/30=1/10 | 1/20+1/20=1/10 |
对于1 ≤ n ≤ 9，方程一共有多少个解？