Let m be an odd integer. Show that the number racm4 + 4m2 + 1116 is an integer.

Here are the solutions to the problems: 1. Let m be an odd integer. Show that the number (m^4 + 4m^2 + 11)/(16) is an integer. Step 1: Express m in terms of an integer k. Since m is an odd integer, we can write m = 2k + 1 for some integer k. Step 2: Find an expression for m^2. m^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1 Since k(k+1) is always an even integer (the product of two consecutive integers), let k(k+1) = 2j for some integer j. Then, m^2 = 4(2j) + 1 = 8j + 1. Step 3: Substitute m^2 = 8j+1 into the given expression's numerator. The numerator is m^4 + 4m^2 + 11. m^4 + 4m^2 + 11 = (m^2)^2 + 4m^2 + 11 = (8j+1)^2 + 4(8j+1) + 11 = (64j^2 + 16j + 1) + (32j + 4) + 11 = 64j^2 + 16j + 32j + 1 + 4 + 11 = 64j^2 + 48j + 16 Step 4: Factor out 16 from the numerator. 64j^2 + 48j + 16 = 16(4j^2 + 3j + 1) Step 5: Substitute the factored numerator back into the original expression. (m^4 + 4m^2 + 11)/(16) = (16(4j^2 + 3j + 1))/(16) = 4j^2 + 3j + 1 Since j is an integer, 4j^2 + 3j + 1 is also an integer. Therefore, the number (m^4 + 4m^2 + 11)/(16) is an integer. 2. If a, b, and c are positive real numbers, then show that: sqrt((b)/(a)) · sqrt((c)/(b)) · sqrt((a)/(c)) = 1 Step 1: Combine the terms under a single square root. Since a, b, c are positive real numbers, we can use the property sqrt(x) · sqrt(y) · sqrt(z) = sqrt(xyz). sqrt((b)/(a)) · sqrt((c)/(b)) · sqrt((a)/(c)) = sqrt((b)/(a) · (c)/(b) · (a)/(c)) Step 2: Simplify the expression inside the square root. sqrt((b · c · a)/(a · b · c)) Cancel out the common terms a, b, c from the numerator and denominator. = sqrt(1) Step 3: Evaluate the square root. sqrt(1) = 1 Thus, sqrt((b)/(a)) · sqrt((c)/(b)) · sqrt((a)/(c)) = 1. 3. Suppose sqrt(n) = (3)/(2), find -sqrt(n) Step 1: Substitute the given value of sqrt(n). We are given that sqrt(n) = (3)/(2). We need to find the value of -sqrt(n). -sqrt(n) = - ((3)/(2)) Step 2: Simplify the expression. -sqrt(n) = -(3)/(2) The value of -sqrt(n) is -(3)/(2). 4. Let a and b be positive real numbers. Show that (a)/(b) + (b)/(a) 2 Step 1: Apply the AM-GM inequality. For any two positive real numbers x and y, the Arithmetic Mean-Geometric Mean (AM-GM) inequality states that $\