32 divides n-6 and 64 does not divide n-6.find the highest power of 2 dividing n+2 and n+6?
To find the highest power of 2 that divides both n+2 and n+6, we need to analyze the given information.
First, we're told that 32 divides n-6. This means that n-6 is divisible by 2^5 (since 32 = 2^5).
Second, we're told that 64 does not divide n-6. Thus, we know that n-6 is not divisible by 2^6 (since 64 = 2^6).
Now, let's consider n+2 and n+6.
We can rewrite n+2 as (n-6) + 8. Since (n-6) is divisible by 2^5, and 8 = 2^3, n+2 will be divisible by 2^5 * 2^3. Therefore, n+2 is divisible by 2^8 (since 2^5 * 2^3 = 2^8).
Next, we can rewrite n+6 as (n-6) + 12. Again, since (n-6) is divisible by 2^5, and 12 = 2^2 * 3, n+6 will be divisible by 2^5 * (2^2 * 3). This simplifies to 2^5 * 2^2 * 3 = 2^7 * 3.
Therefore, the highest power of 2 that divides both n+2 and n+6 is 2^7.
Note: It's important to understand that this explanation works based on the assumption that the given information is accurate and complete.