Step 1: Understanding gauche interactions.
Gauche interactions occur when two substituents on a butane molecule are in a staggered conformation and are within 60° of each other. This leads to steric strain, known as gauche strain.
Step 2: Analyze the compound.
In this compound, there is one gauche-butane interaction between the two methyl groups (CH\(_3\)) attached to the butane backbone. The methyl groups are in a staggered arrangement, 60° apart. This interaction contributes to steric strain.
Step 3: Conclusion.
Thus, the number of gauche-butane interactions in this compound is 1.