jmc

Let $x$ be Tracy's starting number of candies. After eating $\frac{1}{3}$ of them, she had $\frac{2}{3}x$ left. Since $\frac{2}{3}x$ is an integer, $x$ is divisible by 3. After giving $\frac{1}{4}$ of this to Rachel, she had $\frac{3}{4}$ of $\frac{2}{3}x$ left, for a total of $\frac{3}{4}\cdot \frac{2}{3}x=\frac{1}{2}x$. Since $\frac{1}{2}x$ is an integer, $x$ is divisible by 2. Since $x$ is divisible by both 2 and 3, it is divisible by 6.

After Tracy and her mom each ate 15 candies (they ate a total of 30), Tracy had $\frac{1}{2}x-30$ candies left. After her brother took 1 to 5 candies, Tracy was left with 3. This means Tracy had 4 to 8 candies before her brother took some candies. Hence, $$4\le \frac{1}{2}x-30\le 8\Rightarrow 34\le \frac{1}{2}x\le 38\Rightarrow 68\le x\le 76.$$Since $x$ is divisible by 6, and the only multiple of 6 in the above range is 72, we have $x=\boxed{72}$.