“ 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 trial-and-error 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
To find the smallest value for y + z, we need to analyze the given information.
From the first statement, we know that 5y is the square of a number. Therefore, 5y must be a perfect square. This means that y must be a multiple of 5. Let's say y = 5k, where k is an integer.
Next, from the second statement, we know that 75yz is the cube of a number. Therefore, 75yz must be a perfect cube. Since we already have y = 5k, we can substitute it into the equation to get 75(5k)z = 375kz. To make it a perfect cube, there must be a factor of 375 that is a perfect cube. The prime factorization of 375 is 3 * 5 * 5 * 5.
Since y is a multiple of 5, z must contain at least one factor of 5. Now, we need to consider the smallest value for z, which consists of factors other than 5. The remaining factors of 375 are 3 and 5. Let's assume z contains one factor of 3 and one factor of 5.
So, the prime factorization of z will be z = 3 * 5 = 15.
Finally, we can find the smallest value of y + z by substituting y = 5k and z = 15 into the equation:
y + z = 5k + 15
Since we are looking for the smallest possible value, we need to find the smallest value of k that makes 5k + 15 a positive integer greater than 1.
When k = 1, y + z = 5(1) + 15 = 5 + 15 = 20.
Thus, the smallest value for y + z is 20.