Try to factor it!

Number Theory Level 4

x y − 5 y − 2 x = 13 \large xy - 5y - 2x = 13

Given that x x and y y are integers satisfying the equation above, find the sum of all possible values of the product x y xy .

Hints :

  • 13 = 23 − 10 13= 23 - 10 .
  • 23 is prime .


The answer is 132.

View wiki
Ismet Cosic by ismet cosic , 21, Bosnia-Herzego…

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Rishabh Jain
Jul 2, 2016

Relevant wiki: Quadratic Diophantine Equations - Solve by Simon's Favorite Factoring Trick

x y − 5 y − 2 x + 10 = 23 xy-5y-2x+10=23 ⟹ ( x − 5 ) ( y − 2 ) = 23 = 23 ⋅ 1 = ( − 23 ) ⋅ ( − 1 ) = 1 ⋅ 23 = ( − 1 ) ⋅ ( − 23 ) \implies (x-5)(y-2)=23=23\cdot 1=(-23)\cdot (-1)=1\cdot 23=(-1)\cdot (-23) ⟹ ( x , y ) ≡ ( 28 , 3 ) ; ( − 18 , 1 ) ; ( 6 , 25 ) ; ( 4 , − 21 ) \implies (x,y)\equiv (28,3);(-18,1);(6,25);(4,-21)

∴ 84 + ( − 18 ) + 150 + ( − 84 ) = 132 \therefore 84+(-18)+150+(-84)=\boxed{132}

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Very nice!

ismet cosic - 4 years, 11 months ago

Log in to reply

Thanks... :-)

Rishabh Jain - 4 years, 11 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...