Question

What is the smallest positive integer $n$ such that $g^n = e$?

• A. 4
• B. 5
• C. 2
• D. 3
• A. e
• B. f
• C. g
• D. h
• A. e
• B. f
• C. g
• D. h

Answer 1

The Correct Option is D

From the first table, we can compute the powers of $g$. We know that: \[ g^1 = g, \quad g^2 = g \circ g = f, \quad g^3 = g \circ g \circ g = e \] Thus, the smallest positive integer $n$ such that $g^n = e$ is 3.

Answer 2

From the table for $\circ$ operation, we can simplify step by step: \[ f \circ f = g, \quad f \circ (f \circ f) = f \circ g = h \] Now compute: \[ f \circ f \circ f = f \circ h = g \] Thus, $f \circ f \circ f \circ (f \circ (f \circ f)) = f$.

Answer 3

We simplify the expression step by step: \[ g \circ g = f \quad \text{(from the first table)} \] Now we compute: \[ f \circ f = g \quad \text{and then} \quad a^{10} \circ g = e \] Thus, the entire expression simplifies to $e$.