6 x 3 is divisible by (6 + 3)

Number Theory Level 3

How many unordered pairs $(m,n)$ are there, such that $m$ and $n$ are coprime positive integers less than 100, and $\dfrac{mn}{m+n}$ is an integer?

The answer is 0.

1 solution

Kenny Lau
Aug 8, 2015

Since $m$ and $n$ are coprime:
1: $m+n$ and $m$ are also coprime.
2: $m+n$ and $n$ are also coprime.
Combining 1 and 2: $m+n$ and $mn$ are also coprime.
For any two coprime integers $a$ and $b$ , $a$ is divisible by $b$ iff $b=1$ , i.e. $m+n=1$ .
This is impossible as $m$ and $n$ are positive.

Very nice solution!

Adarsh Kumar - 5 years, 10 months ago

Thank you!

Kenny Lau - 5 years, 10 months ago

It is superb! Far easier than mine.

A Former Brilliant Member - 5 years, 10 months ago

I would like to hear yours! :)

Kenny Lau - 5 years, 10 months ago

6 x 3 is divisible by (6 + 3)

The answer is 0.

1 solution

0 pending reports