Question

已知正整数a,b,c满足a＞b＞c,且ab+bc+ca=abc,求所有符合条件的 a,b,c

Answer 1

ab+bc+ca=abc
同时除以 abc

1/c + 1/a + 1/b = 1
由于
a > b > c
所以
1/a<1/b<1/c
1/c + 1/a + 1/b < 3/c
1 < 3/c
c < 3
c = 1 or 2
因为 c〉1
所以 c = 2

1/2 + 1/a + 1/b = 1

1/a + 1/b = 1/2
1/a < 1/b
2/b > 1/2
2 > 1/2b
4 > b
b = 3 or 2 or 1
距题意
b = 3

1/c + 1/a + 1/b = 1
1/2 + 1/3 + 1/a =1
1/a = 1/6
a = 6
所以 a =6 b =3 c =2