answer:To solve this problem, we need to calculate the Euclidean distance between two points (P(-1, 2, -3)) and (Q(3, -2, -1)) in 3-dimensional space. The Euclidean distance between two points ((x_1, y_1, z_1)) and ((x_2, y_2, z_2)) is given by the formula:[d = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}]Let's break down the problem and implement this formula in Python using the SymPy library.1. Define the coordinates of points (P) and (Q).2. Calculate the differences in each coordinate.3. Compute the squared differences, sum them up, and take the square root to get the distance.Here is the Python code to calculate the distance:```pythonimport sympy as sp# Define coordinates for points P and QP = (-1, 2, -3)Q = (3, -2, -1)# Calculate the differences in each coordinatex_diff = Q[0] - P[0]y_diff = Q[1] - P[1]z_diff = Q[2] - P[2]# Calculate the squared differencesdistance_squared = x_diff2 + y_diff2 + z_diff2# Calculate the distancedistance = sp.sqrt(distance_squared)print(distance)``````output6```The Euclidean distance between the points (P(-1, 2, -3)) and (Q(3, -2, -1)) is (boxed{6}).

answer:To solve for the modulus of the complex number (z) given the equation (z(2 - 3i) = 6 + 4i), we can follow these steps:1. Express (z) in terms of its real and imaginary parts.2. Substitute (z = a + bi) into the given equation.3. Separate the real and imaginary parts to form a system of linear equations.4. Solve the system of equations to find (a) and (b).5. Calculate the modulus of (z) using the formula (|z| = sqrt{a^2 + b^2}).Let's implement these steps using Python and SymPy:```pythonimport sympy as sp# Define the symbolsa, b = sp.symbols('a b', real=True)# Define z and the given equationz = a + b*sp.Ilhs = z * (2 - 3*sp.I)rhs = 6 + 4*sp.I# Separate the real and imaginary partsequations = sp.Eq(lhs, rhs)real_eq = sp.Eq(sp.re(lhs), sp.re(rhs))imag_eq = sp.Eq(sp.im(lhs), sp.im(rhs))# Solve the system of equationssolution = sp.solve([real_eq, imag_eq], (a, b))# Extract the real and imaginary parts of za_val = solution[a]b_val = solution[b]# Calculate the modulus of zmodulus_z = sp.sqrt(a_val2 + b_val2)print(modulus_z)``````output2```The modulus of the complex number (z) satisfying the equation (z(2 - 3i) = 6 + 4i) is (boxed{2}).

answer:Given the problem, we need to find a positive integer ( n ) less than 2006 such that ( 2006n ) is a multiple of ( 2006 + n ).This translates to the requirement that:[ frac{2006n}{2006 + n} ]must be an integer.We can reformulate this problem using mathematical reasoning and an iterative code approach to identify the correct ( n ). Let's denote ( k ) as the integer result, i.e.,[ 2006n = k(2006 + n) ]We need to test for each ( n ) from 1 to 2005 whether it satisfies the above condition.Let's implement the code to find the correct ( n ).```python# Let's iterate over the possible values of n from 1 to 2005.# We check if (2006 * n) / (2006 + n) is an integerdef find_n(): for n in range(1, 2006): if (2006 * n) % (2006 + n) == 0: return nn = find_n()print(n)``````output1475```The positive integer ( n ) less than 2006 such that ( 2006n ) is a multiple of ( 2006 + n ) is:(boxed{1475})

answer:To solve the problem, we need to find the number of integer values of ( n ) that satisfy the equation ((n^2 - n - 1)^{n+2} = 1).To break this down:1. The expression ((n^2 - n - 1)^{n+2}) can equal 1 in a few cases: - Case 1: (n^2 - n - 1 = 1) (since (1^{n+2} = 1)) - Case 2: (n^2 - n - 1 = -1) and (n+2) is even (since ((-1)^{text{even number}} = 1)) - Case 3: (n+2 = 0) (since any non-zero number raised to the power of 0 is 1)Let's implement this step-by-step in Python using sympy to solve for (n).```pythonimport sympy as sp# Define the variablen = sp.symbols('n')# Case 1: n^2 - n - 1 = 1eq1 = sp.Eq(n2 - n - 1, 1)solutions_case_1 = sp.solve(eq1, n)# Case 2: n^2 - n - 1 = -1 and n + 2 is eveneq2 = sp.Eq(n2 - n - 1, -1)solutions_case_2 = sp.solve(eq2, n)solutions_case_2 = [sol for sol in solutions_case_2 if (sol + 2) % 2 == 0]# Case 3: n + 2 = 0eq3 = sp.Eq(n + 2, 0)solutions_case_3 = sp.solve(eq3, n)# Combine all solutions and find unique valuesall_solutions = set(solutions_case_1 + solutions_case_2 + solutions_case_3)# Output the number of unique integer solutionsprint(len(all_solutions))``````output4```The number of integer values of ( n ) that satisfy the equation ((n^2 - n - 1)^{n+2} = 1) is (boxed{4}).

answer:To determine how many positive integers ( n ), where ( 10 leq n leq 100 ), satisfy the condition that (frac{n^2 - 9}{n^2 - 7}) is in its lowest terms, we need to check each integer in this range and verify if the fraction is in reduced form.A fraction is in its lowest terms if the numerator and the denominator are coprime (i.e., their greatest common divisor (GCD) is 1). Therefore, for each ( n ) in the given range, we need to check whether the GCD of (n^2 - 9) and (n^2 - 7) is 1.Let's implement this in Python and find the final answer.```pythonimport math # Function to check if the fraction is in reduced form def is_reduced_form(n): numerator = n2 - 9 denominator = n2 - 7 return math.gcd(numerator, denominator) == 1 count = 0 # Iterate through the range 10 to 100 for n in range(10, 101): if is_reduced_form(n): count += 1 print(count)``````output46```The number of positive integers ( n ), where ( 10 leq n leq 100 ), such that (frac{n^2 - 9}{n^2 - 7}) is a fraction in its lowest terms, is (boxed{46}).

answer:We need to find a four-digit number which satisfies the following conditions:1. All digits are distinct.2. The sum of the digits is 16.3. One digit is twice another digit.4. One digit is three times another digit.Let's denote the four digits by (a), (b), (c), and (d), and (N = 1000a + 100b + 10c + d) by the four-digit number, where (a neq b neq c neq d).We will use Python to iterate over all combinations of four distinct digits to find the number that satisfies all the conditions mentioned.1. The digits must be between 0 and 9 inclusive.2. The sum (a + b + c + d) must be 16.3. At least one of the digits must be twice another digit.4. At least one of the digits must be three times another digit.We will run a loop over all possible digit combinations to check these conditions and store the smallest valid number. Let's start with the implementation:```pythonimport itertoolsdef find_smallest_number(): smallest_number = float('inf') digits = range(10) for comb in itertools.permutations(digits, 4): a, b, c, d = comb if sum(comb) == 16: digits_set = set(comb) twice_condition = any(2*x in digits_set for x in digits_set) thrice_condition = any(3*x in digits_set for x in digits_set) if twice_condition and thrice_condition: number = 1000*a + 100*b + 10*c + d if number < smallest_number: smallest_number = number return smallest_numbersmallest_number = find_smallest_number()print(smallest_number)``````output169```The smallest four-digit number that satisfies the given conditions is (169). However, since 169 is not a four-digit number, there may be an issue in our code logic. We need to ensure each digit meets all specified conditions. Let's adjust the code to ensure it properly checks for four-digit numbers and their conditions.```pythondef find_smallest_valid_four_digit_number(): smallest_number = float('inf') digits = range(10) for comb in itertools.permutations(digits, 4): a, b, c, d = comb if sum(comb) == 16: digits_set = set(comb) twice_condition = any(2*x in digits_set and x != 0 for x in digits_set) thrice_condition = any(3*x in digits_set and x != 0 for x in digits_set) if twice_condition and thrice_condition: number = 1000*a + 100*b + 10*c + d if 1000 <= number < smallest_number: smallest_number = number return smallest_numbersmallest_number = find_smallest_valid_four_digit_number()print(smallest_number)``````output1267```The smallest four-digit number that satisfies the given conditions is (boxed{1267}).