To find the GCD (Greatest Common Divisor) of two numbers when their LCM (Least Common Multiple) is given as 1024 and one of the numbers is a prime number, we can follow these steps:
### Step 1: Understand the relationship between LCM and GCD
The relationship between LCM and GCD of two numbers \(a\) and \(b\) is given by the formula:
\[
\text{LCM}(a, b) \times \text{GCD}(a, b) = a \times b
\]
### Step 2: Factorize the LCM
We need to factorize the LCM, which is given as 1024.
\[
1024 = 2^{10}
\]
This means that 1024 is made up of the prime number 2 multiplied by itself 10 times.
### Step 3: Identify the prime number
Since one of the numbers is a prime number, and the only prime factor of 1024 is 2, we can conclude that one of the numbers must be 2.
### Step 4: Determine the second number
Let’s denote the two numbers as \(a\) and \(b\). We have:
- \(a = 2\) (the prime number)
- \(b\) is unknown.
Using the LCM formula:
\[
\text{LCM}(2, b) = 1024
\]
Since 2 is a factor of 1024, we can express \(b\) in terms of 1024:
\[
b = \frac{\text{LCM}(2, b) \times \text{GCD}(2, b)}{2}
\]
### Step 5: Calculate GCD
Now, we can use the relationship:
\[
\text{GCD}(2, b) = \frac{\text{LCM}(2, b)}{b}
\]
Since \(b\) must also be a power of 2 (to maintain the LCM as a power of 2), we can express \(b\) as \(2^n\) for some integer \(n\).
### Step 6: Solve for \(n\)
Using the LCM of powers of 2:
\[
\text{LCM}(2, 2^n) = 2^{\max(1, n)} = 1024 = 2^{10}
\]
This means \(n\) must be 10, so:
\[
b = 2^{10} = 1024
\]
### Step 7: Calculate GCD
Now we can find the GCD:
\[
\text{GCD}(2, 1024) = 2
\]
### Final Answer
Thus, the GCD of the two numbers is:
\[
\text{GCD} = 2
\]
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