Key Concepts and Formulas

• The number of subsets of a set with $k$ elements is $2^k$.
• Solving exponential equations involving powers of 2 often benefits from prime factorization and factoring out common terms.

Step-by-Step Solution

Step 1: Express the number of subsets for sets A and B. Let set A have $m$ elements. The total number of subsets of A is $2^m$. Let set B have $n$ elements. The total number of subsets of B is $2^n$.

Step 2: Formulate an equation based on the given information. The problem states that the total number of subsets of A is 112 more than the total number of subsets of B. This can be written as: $(\text{Number of subsets of A}) = (\text{Number of subsets of B}) + 112$ Substituting the expressions from Step 1, we get: $2^m = 2^n + 112$ Rearranging the equation to isolate the terms with powers of 2: $2^m - 2^n = 112$

Step 3: Solve the exponential equation for $m$ and $n$. Since $2^m - 2^n = 112$ is positive, we know that $2^m > 2^n$, which implies $m > n$. This allows us to factor out the smaller power of 2, which is $2^n$: $2^n (2^{m-n} - 1) = 112$ Now, we find the prime factorization of 112: $112 = 2 \times 56 = 2 \times 2 \times 28 = 2 \times 2 \times 2 \times 14 = 2 \times 2 \times 2 \times 2 \times 7$ So, $112 = 2^4 \times 7$. The equation becomes: $2^n (2^{m-n} - 1) = 2^4 \times 7$ We equate the power of 2 terms and the odd terms on both sides. The term $(2^{m-n} - 1)$ must be odd because $m-n > 0$, making $2^{m-n}$ an even number, and an even number minus 1 is always odd. Equating the powers of 2: $2^n = 2^4$ This gives us $n = 4$. Equating the odd terms: $2^{m-n} - 1 = 7$ Add 1 to both sides: $2^{m-n} = 8$ Since $8 = 2^3$, we have: $2^{m-n} = 2^3$ This gives us $m-n = 3$.

Step 4: Determine the values of $m$ and $n$ and calculate $m \cdot n$. We have a system of two linear equations:

1. $n = 4$
2. $m - n = 3$ Substitute the value of $n$ from the first equation into the second equation: $m - 4 = 3$ $m = 3 + 4$ $m = 7$ So, $m=7$ and $n=4$. The question asks for the value of $m \cdot n$: $m \cdot n = 7 \times 4$ $m \cdot n = 28$

Common Mistakes & Tips

• Incorrect Subset Formula: Remember the number of subsets is $2^k$, not $k^2$ or any other variation.
• Factoring Strategy: When solving $2^m - 2^n = K$, always factor out $2^n$ (assuming $m > n$). This is crucial for separating the powers of 2 from the odd factors.
• Integer Constraints: $m$ and $n$ must be non-negative integers as they represent the number of elements in a set.

Summary

The problem requires finding two integers $m$ and $n$ such that the difference between the number of subsets of two sets ($2^m$ and $2^n$) is 112. By setting up the equation $2^m - 2^n = 112$, factoring out $2^n$, and using the prime factorization of 112, we uniquely determine $n=4$ and $m=7$. Consequently, the product $m \cdot n$ is calculated.

The final answer is $\boxed{28}$.