Question

[Trần Thị Hà Thương]

Tim so tu nhien abc biet 1/a + 1/b + 1/c = 1

Answer 1

[✓ ℍɠŞ_ŦƦùM $₦G ✓]

giả sử a\(\le\)b\(\le\)c

=>\(\frac{1}{a}\ge\frac{1}{b}\)\(\ge\frac{1}{c}\)

=>\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1=\frac{3}{3}<\frac{1}{a}+\frac{1}{a}+\frac{1}{a}=\frac{3}{a}\)

=>\(a\le3\)

=>a=2;3 vì a khác 0 và nếu a=1 thì \(\frac{1}{a}+\frac{1}{b}=0\)(vô lí)

xét a=2\(\Rightarrow\frac{1}{b}+\frac{1}{c}=1-\frac{1}{2}=\frac{1}{2}=\frac{2}{4}\le\frac{1}{b}+\frac{1}{b}=\frac{2}{b}\)

\(\Rightarrow b\le4\)

=>b=2;3;4

xét b=2=>\(\frac{1}{c}=0\)(vô lí)

xét b=3 =>\(\frac{1}{c}=\frac{1}{6}\Rightarrow c=6\)

xét b=4 =>\(\frac{1}{c}=\frac{1}{4}\Rightarrow c=4\)

xét a=3 =>\(\frac{1}{b}+\frac{1}{c}=\frac{2}{3}\le\frac{1}{b}+\frac{1}{b}=\frac{2}{b}\)

=>\(b\le3\)

xét b=2 =>\(\frac{1}{c}=\frac{1}{6}\Rightarrow c=6\)

xét b=3 =>\(\frac{1}{c}=\frac{1}{3}\Rightarrow c=3\)

vậy (a;b;c)=(2;3;4);(2;4;3);(3;2;4);(3;4;2);(4;2;3);(4;3;2);(3;2;6);(3;6;2);(2;3;6);(2;6;3);(6;3;2);(6;2;3);(3;3;3)