If n(A) = a, n(B) = b, and the number of subsets of A exceeds subsets of B by 3072, then find the values of a and b.

Answer

To solve the problem step by step, we need to find the values of \( a \) and \( b \) given that the number of subsets of set \( A \) exceeds the number of subsets of set \( B \) by 3072.

### Step 1: Understand the number of subsets
The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, for sets \( A \) and \( B \):
- The number of subsets of \( A \) is \( 2^a \)
- The number of subsets of \( B \) is \( 2^b \)

### Step 2: Set up the equation
According to the problem, the number of subsets of \( A \) exceeds the number of subsets of \( B \) by 3072. This gives us the equation:
\[ 2^a = 2^b + 3072 \]

### Step 3: Rearranging the equation
Rearranging the equation, we have:
\[ 2^a - 2^b = 3072 \]

### Step 4: Factor out \( 2^b \)
We can factor out \( 2^b \) from the left side:
\[ 2^b (2^{a-b} - 1) = 3072 \]

### Step 5: Factor 3072
Next, we need to factor 3072 to find possible values for \( 2^b \) and \( 2^{a-b} - 1 \).

The prime factorization of 3072 is:
\[ 3072 = 2^{10} \times 3 \]

### Step 6: Set up possible values for \( 2^b \)
From the equation \( 2^b (2^{a-b} - 1) = 3072 \), we can see that \( 2^b \) must be a power of 2. The possible values for \( 2^b \) can be:
- \( 1 \) (which is \( 2^0 \))
- \( 2 \) (which is \( 2^1 \))
- \( 4 \) (which is \( 2^2 \))
- \( 8 \) (which is \( 2^3 \))
- \( 16 \) (which is \( 2^4 \))
- \( 32 \) (which is \( 2^5 \))
- \( 64 \) (which is \( 2^6 \))
- \( 128 \) (which is \( 2^7 \))
- \( 256 \) (which is \( 2^8 \))
- \( 512 \) (which is \( 2^9 \))
- \( 1024 \) (which is \( 2^{10} \))

### Step 7: Calculate corresponding \( 2^{a-b} - 1 \)
For each value of \( 2^b \), we can calculate \( 2^{a-b} - 1 \) by dividing 3072 by \( 2^b \) and adding 1.

1. If \( 2^b = 1024 \):
\[ 2^{a-b} - 1 = \frac{3072}{1024} = 3 \implies 2^{a-b} = 4 \implies a - b = 2 \]
Here, \( b = 10 \) and \( a = b + 2 = 12 \).

### Step 8: Verify the values
Now, we can verify:
- Number of subsets of \( A \): \( 2^{12} = 4096 \)
- Number of subsets of \( B \): \( 2^{10} = 1024 \)
- Difference: \( 4096 - 1024 = 3072 \)

Thus, the values of \( a \) and \( b \) are:
\[ \boxed{12} \text{ and } \boxed{10} \]