1/x+1/y=1/z where x=12 and y,z positive integer how many possible values of y…

Key idea: express y in terms of z and use divisibility constraints.

• Substitute x = 12 into the equation 1/x + 1/y = 1/z to get 1/12 + 1/y = 1/z.

• Rearrange to 1/y = (12 - z)/(12z), so y = 12z/(12 - z).

• Because y must be positive, 12 - z > 0, so z is an integer from 1 to 11.

• Let k = 12 - z (so 1 ≤ k ≤ 11). Then y = 12(12 - k)/k = 144/k - 12, so y is an integer exactly when k divides 144.

• Find positive divisors of 144 that are less than 12: 1, 2, 3, 4, 6, 8, 9. There are 7 such k.

• For these k values, compute z = 12 - k and y = 144/k - 12, giving the valid pairs (y, z): (132, 11), (60, 10), (36, 9), (24, 8), (12, 6), (6, 4), (4, 3).

• Therefore there are 7 possible (y, z) pairs.

Answer: 7