CAT 2021 Slot 2 QA Question 13: Three positive integers $x, y$ and $z$ are in arithmetic progression. If $y - x 2$ and $xyz = 5(x + y + z)$, then $z - x$ equals

Question

Three positive integers $x, y$ and $z$ are in arithmetic progression. If $y - x > 2$ and $xyz = 5(x + y + z)$, then $z - x$ equals

Accepted Answer

Let the three terms be $y - d$, $y$ and $y + d$, where $d$ is a positive integer. Given $y - x > 2$, we have $d > 2$.

$x=y-d, z=y+d$
$xyz=(y-d)y(y+d)=y(y^2-d^2)$
$x+y+z=3y$

Using $xyz = 5(x + y + z)$,
$y(y^2-d^2)=15y$
$y^2-d^2=15$
$(y-d)(y+d)=15$

The positive factor pairs of 15 are (1, 15) and (3

Three positive integers x,yx, yx,y and zzz are in arithmetic progression. If y−x>2y - x > 2y−x>2 and xyz=5(x+y+z)xyz = 5(x + y + z)xyz=5(x+y+z), then z−xz - xz−x equals