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
✅ Correct. Your solution correctly uses modular arithmetic to show that the numerator is a multiple of 16. Here's a breakdown of your steps: 1. You correctly established that if m is an odd integer, then m^2 1 8. This is because m^2 = (2k+1)^2 = 4k(k+1)+1, and k(k+1) is always even, so 4k(k+1) is a multiple of 8. 2. You correctly deduced that m^4 = (m^2)^2 1^2 1 16. This follows from m^2 1 8, which implies m^2 = 8j+1. Then m^4 = (8j+1)^2 = 64j^2 + 16j + 1 = 16(4j^2+j)+1 1 16. 3. For the term 4m^2: Since m^2 1 8, m^2 can be 1, 9, 17, . If m^2 1 16, then 4m^2 4(1) 4 16. If m^2 9 16, then 4m^2 4(9) = 36 4 16. In both cases, 4m^2 4 16. Your use of 4(1) in the calculation 1+4(1)+11 is effectively correct for the modulo 16 result. 4. You correctly summed the terms modulo 16: m^4 + 4m^2 + 11 1 + 4 + 11 16 m^4 + 4m^2 + 11 16 16 m^4 + 4m^2 + 11 0 16 5. Your conclusion that (m^4 + 4m^2 + 11)/(16) is an integer is correct because the numerator is a multiple of 16.