This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define the number of subsets for each set. Let the first set have M elements. The number of subsets for the first set is 2^M. Let the second set have N elements. The number of subsets for the second set is 2^N. Step 2: Set up the equation based on the problem statement. The number of subsets of the first set is 30 more than the second: 2^M = 2^N + 30 Step 3: Rearrange the equation. 2^M - 2^N = 30 Factor out 2^N: 2^N (2^M-N - 1) = 30 Step 4: Find integer solutions for N and M. We need to find factors of 30 that are powers of 2. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The only power of 2 that is a factor of 30 is 2^1 = 2. So, we must have 2^N = 2. Therefore, N=1. Step 5: Substitute N=1 back into the equation. 2^1 (2^M-1 - 1) = 30 2 (2^M-1 - 1) = 30 Divide both sides by 2: 2^M-1 - 1 = 15 Add 1 to both sides: 2^M-1 = 16 Express 16 as a power of 2: 2^M-1 = 2^4 Equate the exponents: M-1 = 4 M = 4 + 1 M = 5 The values are M=5 and N=1. The final answer is M=5, N=1.