Prove or disprove the statement. For all integers x, y and z, if x|y and x|z, then x|( ) (6𝑦2 − 𝑧2
Question
Prove or disprove the statement.
For all integers x
, y
, and z
, if
x | y
x | z
then
x | (6y² - z²)
.
Solution
To prove the statement, we will follow a structured approach.
1. Break Down the Problem
We need to prove that if divides and (denoted as and ), then must also divide the expression .
2. Relevant Concepts
According to the definition of divisibility, if , then there exists an integer such that: Similarly, if , then: for some integer .
3. Analysis and Detail
Next, we substitute equations (1) and (2) into the expression .
Step 3.1: Substitute and
Using and :
Step 3.2: Expand the terms
Now, we can simplify: Thus:
Step 3.3: Factor the expression
Now, we can factor out :
4. Verify and Summarize
Since is an integer (as both and are integers), we see that:
This means divides . Therefore, since divides , we can conclude:
Final Answer
The statement is true: For all integers , , and , if and , then .
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