Find the error.

A student is adding [tex]\(1 \frac{2}{9}, 3 \frac{1}{3},\)[/tex] and [tex]\(-4 \frac{5}{6}\)[/tex]. The first step the student performs is to find the common denominator of 9, 3, and 6. Find the student's mistake and correct it.

The least common denominator of 9, 3, and 6 is 36 because you can divide 36 by all of these numbers without getting a remainder. The student found a common denominator, but not the least common denominator. The least common denominator of 9, 3, and 6 is 18.

Let's help identify the mistake and correct it.

The student made an error by finding the common multiple instead of finding the least common denominator (LCD) for the denominators 9, 3, and 6.

### The Student's Mistake:
- The student mistakenly identified 36 as a common denominator of 9, 3, and 6. While 36 is a common multiple, it is not the least common denominator.

### Correcting the Mistake:
1. Identify the Least Common Multiple (LCM):
- We need to find the smallest number that 9, 3, and 6 can all divide into without leaving a remainder, which is the Least Common Denominator (LCD).

2. Finding the LCM:
- The LCM of 3 and 6 is 6. (Since 6 is a multiple of both 3 and 6)
- Now, find the LCM of 9 and 6:
- List the multiples of 9: 9, 18, 27, 36...
- List the multiples of 6: 6, 12, 18, 24, 30...
- The smallest common multiple is 18.

3. Conclusion:
- Therefore, the least common denominator of 9, 3, and 6 is 18.

Hence, the student found a common multiple but not the least common denominator. The least common denominator of 9, 3, and 6 is 18.