# Four "Lets" and three sets are perfect with three variables

**Algebra**Level 5

Let \(X = \{ 3a^{2} + 27b^{2} + 5c^{2} - 18ab - 30c + 237 | a, b, c \in Z^{+} \}\).

Let \(x_{0} \in X\) be the lowest value of the set \(X\).

Let \(Y = \{(a, b, c) | 3a^{2} + 27b^{2} + 5c^{2} - 18ab - 30c + 237 = x_{0}\}\)

Let \(Z = \{a + b + c | (a, b, c) \in Y\}\)

Find the member in \(Z\) which has the least value.

**Clarification** : \(Z^{+} \) denote the set of positive integers