- #1
Math9999
Homework Statement
Solve 6x+3=1 in ℤ8.
Homework Equations
None.
The Attempt at a Solution
6x+3=9
6x=6
x=1
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The answers are x=1, or x=5 in the textbook. But how do I get x=5?
Since ##6 \in \mathbb{Z}_8## is no unit, we cannot divide by ##6##. Thus from ##6x=6## we get ##6(x-1)=0## and we need to find all numbers, for which ##6y=0## in ##\mathbb{Z}_8##. Now which multiples of ##6## are divisible by eight?Math9999 said:Homework Statement
Solve 6x+3=1 in ℤ8.
Homework Equations
None.
The Attempt at a Solution
6x+3=9
6x=6
x=1
-----------
The answers x=1, or x=5 in the textbook. But how do I get x=5?
The equation being solved is 6x+3=1 in ℤ8.
In this context, "x=5" means that the value of x that satisfies the equation is equal to 5.
ℤ8 represents the set of integers modulo 8. This means that the possible values of x in this equation are restricted to the numbers 0, 1, 2, 3, 4, 5, 6, and 7.
To solve this equation, we can use basic algebraic principles and operations. First, we can subtract 3 from both sides of the equation to isolate the variable term. This gives us 6x= -2. Then, we can divide both sides by 6 to get the value of x. In this case, x=-2/6 = -1/3. However, since we are working in ℤ8, we must find the value of x that is congruent to -1/3 modulo 8. This gives us x=5, which is our final answer.
Specifying "ℤ8" in this equation is important because it limits the possible values of x to a specific set. In this case, ℤ8 represents a finite set of integers, which allows us to find a unique solution for x. Without specifying "ℤ8", there could be an infinite number of solutions for x, making it difficult to accurately solve the equation.