Quadratics?

Question

Let x, y, z be positive integers. What is the number of positive integers n for which x/n = y/(n+1) = z /(n+2) and x + y + z = 90?

Answer 1

n={1,2,4,5,9,14,29}

n....x...y...z
1...15.30.45
2...20.30.40
4...24.30.36
5...25.30.35
9...27.30.33
14.28.30.32
29.29.30.31

So there are 7 values for n

Answer 2

We have three linear equations in x, y and z... Isolating x in all three of them we get:
(1) x = yn/(n+1)
(2) x = zn/(n+2)
(3) x = 90 - y - z

from (1) and (2) we get
y = z(n+1) / (n+2)

From (2) and (3) we get
y = 90 - z - zn/(n+2)

And from these two we get
z(n+1)/(n+2) = 90 - z - zn/(n+2)
→ z(n+1) = (90-z)(n+2) - zn
→ zn + z = 90n - zn + 180 - 2z - zn
→ z = (90n + 180) / (3n + 3) = 30(n+2)/(n+1).
Plugging this into the above equations ...
→ x = 30n/(n+1) and y = 30.

(n+1) has to be a factor of 30 for x and z to be integers. The solutions are:

n=1 → x=15, y=30, z=45
n=2 → x=20, y=30, z=40
n=4 → x=24, y=30, z=36
n=5 → x=25, y=30, z=35
n=9 → x=27, y=30, z=33
n=14 → x=28, y=30, z=32.
n=29 → x=29, y=30, z=31.

Answer 3

n={1,2}