Number System

Step 1: Start with the given equation. We are given the equation: We are tasked with finding the values of and where both and are natural numbers.
Step 2: Solve for possible values of and . For to hold, must be a power of 2. Let: Substitute into the equation: Thus, for every integer , we can find a corresponding .
Step 3: Identify the valid tuples. Let’s check for small values of :
For , we have and , which gives the tuple .
For , we have and , which gives the tuple .
These are the only valid solutions, as higher values of would give non-natural values for .
Step 4: Conclude the result. Thus, the set contains exactly 2 tuples: and . The cardinality of is: