Question

If x, y, and z are positive integers and 2^x * 3^y * 5^z = 54,000, what is the value of x + y + z

Answer 1

Given:

If x, y, and z are positive integers, then

$2^x\times 3^y\times 5^z=54000$

To find:

The value of $x+y+z$.

Solution:

First we need to find the prime factors of 54000.

$54000=2\times 2\times 2\times 2\times 3\times 3\times 3\times 5\times 5\times 5$

$54000=2^4\times 3^3\times 5^3$        ...(i)

We have,

$54000=2^x\times 3^y\times 5^z$        ...(ii)

On comparing (i) and (ii), we get $x=4,y=3,z=3$.

The sum of $x,y,z$ is:

$x+y+z=4+3+3$

$x+y+z=10$

Therefore, the value of $x+y+z$ is 10.