NMTC Inter Level Problem 11

Algebra Level 4

What is the number of pairs of natural numbers $(x,y)$ which satisfy $\frac{5}{x}+\frac{6}{y}=1?$

8 30 11 5

1 solution

Saurabh Mallik
Aug 28, 2014

According to the question, we need to find pairs of natural numbers not integers.

So, the solutions (pairs) to the equation after solving it are as follows: $\frac{5}{x}+\frac{6}{y}=1$

$x=6$ , $y=36$

$x=7$ , $y=21$

$x=8$ , $y=16$

$x=10$ , $y=12$

$x=11$ , $y=11$

$x=15$ , $y=9$

$x=20$ , $y=8$

$x=35$ , $y=7$

Thus, the total number of pairs: $\boxed{8}$

Due to a bug, I cannot post solution. So, I will post it here:

$\frac{5}{x} + \frac{6}{y} = 1$

$5y + 6x = xy$

$0 = xy -5y - 6x$

Adding 30 both sides,

$30 = xy - 5y - 6x + 30$

$30 = (x-5)(y-6)$

There are 8 ways in which 30 can be expressed as the product of 2 numbers. This can easily be found by finding $\tau(30)$ .

Therefore, there are $\boxed{8}$ solutions.

Kartik Sharma - 6 years, 9 months ago

Excellent solution :)

Krishna Ar - 6 years, 9 months ago

Thanks!!! :)

Kartik Sharma - 6 years, 9 months ago

Hey please tell me about the function u told in last para

Mrigank Krishan - 5 years, 12 months ago

It is the no. of divisors of 30.

Bala vidyadharan - 5 years, 7 months ago

it's not a bug, you marked the answer wrong and then came up with a solution after that , right? it happens with me too.

A Former Brilliant Member - 4 years, 9 months ago