If $$S = \{a \in \mathbb{R} : |2a - 1| = 3[a] + 2\{a\}\}$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$ and $$\{t\}$$ represents the fractional part of $$t$$, then $$72\sum_{a \in S} a$$ is equal to ______

$$a = [a] + \{a\}$$. Let $$[a] = n$$ and $$\{a\} = f$$, where $$f \in [0, 1)$$.

Equation: $$|2(n+f) - 1| = 3n + 2f$$.

Case 1: $$2a - 1 \ge 0 \implies a \ge 1/2$$

$$2n + 2f - 1 = 3n + 2f \implies n = -1$$.

Since $$a = n+f = -1+f$$, the max value is $$<0$$, which contradicts $$a \ge 1/2$$. No solution.

Case 2: $$2a - 1 < 0 \implies a < 1/2$$

$$-(2n + 2f - 1) = 3n + 2f \implies 1 - 2n - 2f = 3n + 2f \implies 4f = 1 - 5n$$.

Since $$0 \le f < 1$$, then $$0 \le \frac{1-5n}{4} < 1$$.

• $$1-5n < 4 \implies -3 < 5n \implies n > -0.6$$

• $$1-5n \ge 0 \implies 5n \le 1 \implies n \le 0.2$$

The only integer $$n$$ in this range is $$n = 0$$.

If $$n = 0$$, $$4f = 1 \implies f = 1/4$$.

$$a = 0 + 1/4 = 1/4$$.

Sum of $$a \in S$$ is $$1/4$$.

$$72 \times (1/4) = 18$$.