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
Let m be an odd integer. We can write m = 2k+1 for some integer k. First, find m^2: m^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1 Since k(k+1) is always an even number (product of two consecutive integers), let k(k+1) = 2j for some integer j. So, m^2 = 4(2j) + 1 = 8j + 1. Next, find m^4: m^4 = (m^2)^2 = (8j+1)^2 = 64j^2 + 16j + 1 = 16(4j^2 + j) + 1 Let P = 4j^2 + j, which is an integer. So, m^4 = 16P + 1. Now, substitute m^4 = 16P+1 and m^2 = 8j+1 into the given expression: (m^4 + 4m^2 + 11)/(16) = ((16P+1) + 4(8j+1) + 11)/(16) = (16P + 1 + 32j + 4 + 11)/(16) = (16P + 32j + 16)/(16) Factor out 16 from the numerator: = (16(P + 2j + 1))/(16) = P + 2j + 1 Since P and j are integers, their sum P+2j+1 is also an integer. Therefore, the number (m^4 + 4m^2 + 11)/(16) is an integer. The expression simplifies to an integer P+2j+1.