# 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

×