1 makes everything beautiful

Number Theory Level 3

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

$xy+x+y=20\\ zy+z+y=6\\ xz+x+z=2$

Given that $x,y,z\neq 0$ , find $z-x-y$

The answer is 10.

1 solution

Patrick Corn
Aug 13, 2014

We get $(x+1)(y+1) = 21$ , $(y+1)(z+1) = 7$ , $(x+1)(z+1) = 3$ . Multiplying the last two and comparing with the first equation shows that $(z+1)^2 = 1$ . Since $z+1 \ne 1$ , we get $z+1 = -1$ and hence $x+1 = -3$ and $y+1 = -7$ . So $z - x - y = -2 - (-4) - (-8) = \fbox{10}$ .

Oh......SFFT :( ...How on ....earth...could I forget this :(

Jayakumar Krishnan - 6 years, 9 months ago

Refrain from random swearing please :)

Joshua Ong - 6 years, 9 months ago

"this word" is mostly accepted as euphemism. It is also not considered swearing at all in many cases..I dont see anything wrong..

Jayakumar Krishnan - 6 years, 9 months ago

@Jayakumar Krishnan – Still, to avoid any such thing on Brilliant, I'm editing your comment.

Satvik Golechha - 6 years, 9 months ago

Lets write the equations as ab+a+b=c ---> a(b+1)=c-b Therefore a=(c-b)/(b+1) plugging this for y gives: x=(20-y)/(y+1) z=(6-y)/(y=1) Using trail and error on the first equation gives us y=6 and x and z are integers Plugging in y=6, tou get z=0 and x=2 Knowing this to be true: 6 2+2+6=20 0 6+0+6=6 2*0+0+2=2 Therefore, z-y-x=0-2-6=-8

M Lin - 6 years, 9 months ago

(z+1)^2=1 has 2 solutions, and z= both -2 and 0 Therefore the question has 2 solutions. Subsituting will get 10 or -8

M Lin - 6 years, 9 months ago

$x,y,z\neq 0$

Tan Wee Kean - 6 years, 9 months ago

yeah too easy question

Karan Shekhawat - 6 years, 6 months ago

1 makes everything beautiful

The answer is 10.

1 solution

0 pending reports