Math
posted by cherryl .
“ y and z are integers greater than 1
5y is the square of a number
75yz is the cube of a number.
The smallest value for y + z is = ?”

The smallest 5y (with y an integer) that is a square is 25 (y = 5). The next smallest is 100 (y = 20). Let's try y=5 and see what z's will work.
Try various integer z's with y = 5
z = 1, 75yz = 375 (not a cube)
z = 2, 75yz = 750 (not a cube)
z = 3, 75yz = 1125 (not a cube)
z = 4, 75yz = 1500 (not a cube)
z = 5, 75yz = 1875 (not a cube)
z = 6, 75yz = 2250 (not a cube)
z = 7 and 8 don't work either, but
For z=9 75yz = 3375 = 15^3
Therefore y = 5, z = 9 and y + z = 14
No lower integers will work. If I had tried y = 20, the sum y+z would have had to exceed 14.
I used a "trial and error" method. I don't see any easier method, but there might be one. 
All we can do is perhaps shorten some of the trialanderror steps that drwls had to make
Start with his determination that if
5y is a perfect square, then y = 5
then if 75yz is a cube
then 75(5)z is a cube
If 375z is a cube
375 = 5 x 5 x 5 x 3 x z
for a perfect cube, each factor should show up 3 times
I see three 5's and one 3, so it looks like we need another pair of 3's, or a 9
so z=9