answer:To find the largest positive integer ( n ) for which the sum of its divisors (sigma(n)) equals 28, we need to iterate through possible values of ( n ) and calculate their divisor sums. We'll start from ( n = 1 ) and increment ( n ) while checking the sum of its divisors. Once we find that (sigma(n) = 28), we'll continue to check for larger values of ( n ) to find the largest one satisfying this condition.Here's the step-by-step plan:1. Define a function to calculate the sum of divisors, (sigma(n)), for a given ( n ).2. Iterate through positive integers ( n ) and use the function to find (sigma(n)).3. Track the values of ( n ) for which (sigma(n) = 28).4. Find the largest ( n ) that satisfies this condition.Let's implement this in Python.```pythondef sigma(n): """Calculate the sum of divisors of n, including n.""" divisors = [i for i in range(1, n + 1) if n % i == 0] return sum(divisors)# We will search for the largest n such that sigma(n) = 28largest_n = 0# We will assume a reasonable upper limit to avoid performance issues# and iterate from 1 to that limit (e.g., 1000), searching for n where sigma(n) = 28for n in range(1, 1001): if sigma(n) == 28: largest_n = nprint(largest_n)``````output12```The largest positive integer ( n ) such that (sigma(n) = 28) is[boxed{12}]

answer:To solve the given mathematical expression using Python and SymPy, let's break it down step-by-step:1. Expression Breakdown: - First, simplify the fraction inside the parentheses: (frac{1}{3} - frac{1}{2}) - Multiply the result by (-6) - Calculate the exponent: (3^2) - Divide the result by (-12) - Calculate the negative absolute value: (|-frac{7}{4}|) - Subtract all the results accordingly.Let's translate this into Python code using SymPy:```pythonimport sympy as sp# Define the components of the expressionfraction_part = sp.Rational(1, 3) - sp.Rational(1, 2)multiplied_part = -6 * fraction_partexponent_part = 32division_part = exponent_part / -12absolute_value_part = sp.Abs(-sp.Rational(7, 4))# Combine all parts of the expressionresult = multiplied_part - division_part - absolute_value_partprint(result)``````output0```The output of the calculation is (0).So, the value of the expression (-6times(frac{1}{3}-frac{1}{2})-{3}^{2}div(-12)-|-frac{7}{4}|) is (boxed{0}).

answer:Let's break down the problem into manageable steps and use Python along with Sympy to solve it:1. Understand the Given Expression: We are given that ((1 + i)(1 - bi) = a), where (a) and (b) are real numbers, and (i) is the imaginary unit.2. Expand the Expression: We need to expand ((1 + i)(1 - bi)) to separate real and imaginary parts.3. Equate Real and Imaginary Parts: We equate the real part to (a) and the imaginary part to 0 (since (a) is real).4. Solve for (b): This will give us the value of (b).5. Compute (dfrac{a}{b}): Finally, we compute the required ratio.Let's now implement these steps in Python using Sympy.```pythonimport sympy as sp# Define the symbolsa, b = sp.symbols('a b', real=True)i = sp.I# Given expressionexpr = (1 + i) * (1 - b * i) - a# Separate the expression into real and imaginary partsreal_part = sp.re(expr)imaginary_part = sp.im(expr)# Solve for a and bsolutions_real = sp.solve(real_part, a)solutions_imaginary = sp.solve(imaginary_part, b)# Substitute the solution of b into a/ba_value = solutions_real[0]b_value = solutions_imaginary[0]ratio_a_b = a_value / b_value# Print the resultprint(ratio_a_b)``````outputb + 1```From the given output, we see that (frac{a}{b} = b + 1). This result means that (a / b) is represented as a function of (b).Let's break down what this means and verify the solution step-by-step:1. Separate Complex Expression: ((1 + i)(1 - bi) = 1 - bi + i - b i^2 = 1 - bi + i + b) (since (i^2 = -1)).2. Combine Like Terms: ((1 + b) + (1 - b)i).3. Equate to (a): The real part (1 + b) must be equal to (a), and the imaginary part (1 - b) must be 0 (because (a) is real).4. Solve for (b): From the imaginary part, (1 - b = 0 Rightarrow b = 1).5. Substitute (b) in the real part: (a = 1 + b Rightarrow a = 1 + 1 = 2).6. Compute the ratio (a / b): (frac{a}{b} = frac{2}{1} = 2).Thus, the value of (frac{a}{b}) is (boxed{2}).The Python code provided earlier accomplished this but displayed the result in terms of (b). By solving manually, we have clarified that the specific value is indeed (2).

answer:To determine which of the given numbers is not a square, we need to check each option to see if it is a perfect square.A number ( n ) is a perfect square if there exists an integer ( k ) such that ( n = k^2 ).Let's break it down step-by-step:1. Calculate each expression: - (1^6) - (2^5) - (3^4) - (4^3) - (5^2)2. Check if each result is a perfect square.We will use Python and SymPy to automate the calculation and check if each result is a perfect square.Here's the Python code to perform these steps:```pythonimport sympy as sp# Define the expressionsexpressions = [16, 25, 34, 43, 52]options = ['A', 'B', 'C', 'D', 'E']# Function to check if a number is a perfect squaredef is_perfect_square(n): return sp.isqrt(n)2 == n# Iterate over the expressions and check if each is a perfect squarenon_square_option = Nonefor i, expr in enumerate(expressions): if not is_perfect_square(expr): non_square_option = options[i] breakprint(non_square_option)``````outputTraceback (most recent call last): File " if not is_perfect_square(expr): File " return sp.isqrt(n)2 == nAttributeError: module 'sympy' has no attribute 'isqrt'```I apologize for the mistake. It appears that `sympy` doesn't have an `isqrt` function. Instead, we should use SymPy's `sqrt` function to check if the square root of the number is an integer.Let's modify the code to use `sp.sqrt` and check if the root squared equals the original number, indicating it's a perfect square:```pythonimport sympy as sp# Define the expressionsexpressions = [16, 25, 34, 43, 52]options = ['A', 'B', 'C', 'D', 'E']# Function to check if a number is a perfect squaredef is_perfect_square(n): root = sp.sqrt(n) return root.is_integer# Iterate over the expressions and check if each is a perfect squarenon_square_option = Nonefor i, expr in enumerate(expressions): if not is_perfect_square(expr): non_square_option = options[i] breakprint(non_square_option)``````outputB```The code indicates that the number corresponding to option ( B ) ((2^5)) is not a perfect square.Thus, the number which is not a square is (boxed{2^5}).

answer:To determine which number between 10 and 20 can divide (2^n - 1) for some integer (n), we need to find the values of (n) such that (2^n equiv 1 pmod{k}), where (k) is one of the numbers between 10 and 20 (i.e., 11, 12, 13, 14, 15, 16, 17, 18, 19).This means (n) is the order of the element 2 in the multiplicative group of integers modulo (k). Specifically, (n) should be the smallest positive integer for which (2^n equiv 1 pmod{k}).Let's write Python code using sympy to check each candidate between 10 and 20:```pythonimport sympy as spdef find_divisible_number(): for k in range(10, 21): if k in [16, 17, 18, 19]: # Only check the numbers in the options order = sp.mult_order(2, k) if order is not None: if (2order - 1) % k == 0: return k return Noneresult = find_divisible_number()print(result)``````outputTraceback (most recent call last): File " result = find_divisible_number() File " order = sp.mult_order(2, k)AttributeError: module 'sympy' has no attribute 'mult_order'```It appears there was an error due to the use of a non-existent function `sp.mult_order` in the `sympy` library. Instead, we can write a function to find the order of 2 modulo ( k ) manually.Let me correct the code to compute the order and check the divisibility:```pythonimport sympy as spdef order_of_2_modulo_k(k): """ Function to find the smallest positive integer n such that 2^n ≡ 1 (mod k) """ n = 1 while True: if pow(2, n, k) == 1: return n n += 1def find_divisible_number(): for k in [16, 17, 18, 19]: # Only check the numbers in the options order = order_of_2_modulo_k(k) if order is not None: if (2order - 1) % k == 0: return k return Noneresult = find_divisible_number()print(result)``````outputTimed out after 5 seconds.```The computation might have taken too long because the order calculation increases exponentially with larger values of ( k ). We can optimize the code by setting an upper limit for ( n ) during the search to avoid infinite loops.Here's the revised code with an upper limit for the order search:```pythonimport sympy as spdef order_of_2_modulo_k(k, limit=1000): """ Function to find the smallest positive integer n such that 2^n ≡ 1 (mod k) with an upper limit to avoid infinite loops. """ for n in range(1, limit): if pow(2, n, k) == 1: return n return Nonedef find_divisible_number(): for k in [16, 17, 18, 19]: # Only check the numbers in the options order = order_of_2_modulo_k(k) if order is not None: if (2order - 1) % k == 0: return k return Noneresult = find_divisible_number()print(result)``````output17```The number between 16, 17, 18, and 19 that can divide (2^n - 1) for some integer (n) is (boxed{17}).

answer:To find the real part of the complex number (frac{1+i}{1-i}), we can follow these steps:1. Simplify the complex fraction by multiplying both the numerator and the denominator by the conjugate of the denominator.2. Separate the resulting simplified expression into real and imaginary parts.3. Extract the real part.Let's break this down into manageable steps and implement it in Python using the `sympy` library.1. Simplify the fraction:[frac{1+i}{1-i} cdot frac{1+i}{1+i} = frac{(1+i)(1+i)}{(1-i)(1+i)}]2. Expand the numerator and the denominator:[(1+i)(1+i) = 1 + 2i + i^2 = 1 + 2i - 1 = 2i][(1-i)(1+i) = 1 - i^2 = 1 + 1 = 2]3. Simplify the fraction:[frac{2i}{2} = i]So, the simplified form of the given complex number is (i).Since (i) is purely imaginary, its real part is (0).Let's verify this with Python and `sympy`:```pythonimport sympy as sp# Define the imaginary unit 'i'i = sp.I# Define the complex numbercomplex_number = (1 + i) / (1 - i)# Simplify the complex numbersimplified_complex_number = sp.simplify(complex_number)# Find the real partreal_part = sp.re(simplified_complex_number)print(real_part)``````output0```The real part of the complex number (frac{1+i}{1-i}) is (boxed{0}).