4.6 Identify and apply the appropriate property of integers to determine the value of \( a, b \) and \( c \) in the equations below. Also, write down the name of the property. \[ a \times(b \times 6)=(-2 \times b) \times c \] \( a \times b=c \) Where \( a, b \) and \( c \) are integers.

To solve the equations and identify the properties of integers involved, let's analyze the given equations step by step. ### Given Equations: 1. \( a \times (b \times 6) = (-2 \times b) \times c \) 2. \( a \times b = c \) ### Step 1: Simplify the First Equation We can use the associative property of multiplication, which states that the way in which numbers are grouped in multiplication does not change the product. This allows us to rewrite the first equation as follows: \[ a \times (b \times 6) = (a \times b) \times 6 \] Now substituting \( c \) from the second equation into the first equation: \[ (a \times b) \times 6 = (-2 \times b) \times c \] Substituting \( c \) gives us: \[ (a \times b) \times 6 = (-2 \times b) \times (a \times b) \] ### Step 2: Factor Out \( a \times b \) Assuming \( a \times b \neq 0 \), we can divide both sides by \( a \times b \): \[ 6 = -2 \] This is a contradiction, which implies that \( a \times b = 0 \). Therefore, either \( a = 0 \) or \( b = 0 \). ### Step 3: Determine Values of \( a, b, c \) 1. If \( a = 0 \): - From the second equation \( 0 \times b = c \) implies \( c = 0 \). - Thus, \( a = 0, b \) can be any integer, and \( c = 0 \). 2. If \( b = 0 \): - From the second equation \( a \times 0 = c \) implies \( c = 0 \). - Thus, \( b = 0, a \) can be any integer, and \( c = 0 \). ### Conclusion The values of \( a, b, c \) can be summarized as: - \( a = 0 \) or \( b = 0 \) (both lead to \( c = 0 \)). ### Properties Used 1. **Associative Property of Multiplication**: \( a \times (b \times c) = (a \times b) \times c \) 2. **Multiplicative Property of Zero**: If \( a \times b = 0 \), then either \( a = 0 \) or \( b = 0 \). Thus, the final values are: - \( a = 0 \) or \( b = 0 \) - \( c = 0 \)