Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m.n is ______.

Solution

Maths sir · 4 steps

Pull out the smaller power of 2: what is left is odd, and that splits 112 in only one way.

1. Count subsets. A set with k elements has 2ᵏ subsets. So 2ᵐ - 2ⁿ = 112, and since the difference is positive, m > n.

   2ⁿ(2ᵐ⁻ⁿ - 1) = 112 = 2⁴ × 7

2. Match the factors. Because m > n, 2ᵐ⁻ⁿ is even, so 2ᵐ⁻ⁿ - 1 is odd. All the factors of 2 in 112 must therefore sit in 2ⁿ, which gives 2ⁿ = 16, so n = 4.
3. Find m. The odd part gives 2ᵐ⁻ⁿ - 1 = 7, so 2ᵐ⁻ⁿ = 8 and m - n = 3. Hence m = 7. Check: 2⁷ - 2⁴ = 128 - 16 = 112.
4. So. m × n = 7 × 4 = 28.

The common slip: 11. Adding the two values instead of multiplying them gives m + n = 11, not m × n = 28.

The idea to keep. For 2ᵐ - 2ⁿ = N, factor out 2ⁿ. The bracket 2ᵐ⁻ⁿ - 1 is odd, so 2ⁿ is exactly the power of 2 in N and the odd part gives m - n.

Answer from the NTA final answer key. Solution by PrepWiser.