Let $a$ and $b$ be two positive even integers. Prove that $(a, b)=2(a / 2, b / 2)$.

Step 1: Understand the properties of even integers An even integer can be defined as any number that is divisible by 2. In other words, any integer $n$ is even if there exists an integer $k$ such that $n=2k$ . Given that $a$ and $b$ are both even integers, we can say that $a=2x$ and $b=2y$ for some integers $x$ and $y$ . Step 2: Rewrite the GCD statement using properties of even integers We are asked to show that $(a,b)=2(a/2,b/2)$ . Using our definitions from step 1, we can rewrite the right side of this statement using $x$ and $y$ . We get: $2(a/2,b/2)=2(x,y)$ . Step 3: Find the GCD using properties of even integers Since $a$ and $b$ are both even, they share a common factor of 2. Therefore, their GCD must also be divisible by 2. So, $(a,b)=2k$ for some integer $k$ . Step 4: Check if the GCD of the halved integers is equal to the divisor We divided both even integers $a$ and $b$ by 2, obtaining the integers $x$ and $y$ . Now we need to find the relationship between the GCD of $x$ and $y$ and the GCD of $a$ and $b$ . Recall that $(a,b)=2k$ . Since $a=2x$ and $b=2y$ , we can substitute these values into the GCD equation: $(2x,2y)=2k$ . Now, since both $2x$ and $2y$ have a common factor of 2, we can divide both sides of the equation by 2: $(x,y)=k$ . Step 5: Combine the results and verify the proof From step 3, we have $(a,b)=2k$ , and from step 4, we have $(x,y)=k$ . Now, remembering that the original problem statement is $(a,b)=2(a/2,b/2)$ , we have: $(a,b)=2(a/2,b/2)=2(x,y)=2k=2(a,b)/2$ . This confirms that the given formula works and simplifies the GCD of two positive even integers.