1998 UK Mathematical Olympiad

Algebra Level 3

Let $x,y,z$ be positive integers such that:

$\frac{1}{x}- \frac{1}{y} = \frac{1}{z}$

If $h$ is the greatest common divisor of $x,y,z$ ,

then $hxyz$ and $h(y-x)$ are not perfect squares.

Is this statement true or false?

False True

2 solutions

Vishal Kumar
Feb 16, 2018

Hint- we can write difference of any two numbers as a multiple of their HCF. Therefore, y-x=kh for some integer k And khxyz=(xy)^2 So, hxyz=(xy)^2/k but since I can be any integer not necessarily a square no. Therefore hxyz could not be perfect square all time.

Thank you for sharing your ideas.

Hana Wehbi - 3 years, 3 months ago

Chan Tin Ping
Feb 20, 2018

Let $x=2,y=3,z=6$ , which $\frac{1}{2}-\frac{1}{3}=\frac{1}{6}$ . We can get their greatest common divisor, $h=1$ .

$hxyz=36=6^2$ $h(y-x)=1(3-2)=1^2$

As there exists a case which contradict the question, the answer is $\large No$ .

Thank you for sharing your solution but l think if we solved it using general variables, it would be better. It would appeal more convincing, but nice solution anyhow.

Hana Wehbi - 3 years, 3 months ago

1998 UK Mathematical Olympiad

2 solutions

0 pending reports