## Solving the diophantine equation ^{1}/_{a}+^{1}/_{b}= ^{p}/_{10n}

### Problem 157

Published on Friday, 1st June 2007, 06:00 pm; Solved by 1703; Difficulty rating: 65%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?