Find the Vault code puzzle

Find the Vault code puzzle

Please find the Vault code consisting of 5 different numbers … (1-9)
1. The sum of the first digit and the sum of the third digit is 10
2. The first digit is a square of the fifth digit
3. The fourth digit is equal to the second digit plus 1
4. The sum of the digits is 22

What is the Vault code?

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Let the number be VWXYZ

Conditions:

(1) V + X = 10
(2) V = Z^2 => V = 4 / 9 and Z = 2 / 3 (V = 1 => Z = 1 which is NOT allowed.)
(3) Y = W + 1
(4) V + W + X + Y + Z = 22

Applying (1) and (3) in (4) => 10 + 2W + 1 + Z = 22

=> 2W + Z = 11 (5)

Exploring possible values from (2):

(a) If V = 4, Z = 2 then (5) is 2W = 9 which is Impossible.
(b) If V = 9, Z = 3 then X = 1, W = 4, Y = 5.

Hence the code is 94153.