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

• A. 8
• B. 10
• C. 14
• D. 12

Answer 1

Answer: (C)

14

Answer 2

Given that x, y, and z are in an arithmetic progression and that $xyz = 5(x + y + z),$ we can start by simplifying the equation:

$xyz = 5(3y)$ because x, y, and z are in an arithmetic progression.

This leads to xz = 15.

Now, we can explore the possible combinations for x and z, which are (3, 5) and (1, 15).

We also know that $z - x > 4$ since $y - x > 2.$

Therefore, when we consider the combination (1, 15), we have:

$z - x = 15 - 1 = 14.$

Hence, $z - x$ equals 14.