answer:To solve the problem, we need to find the greatest common factor (GCF) of 3 and 6, and the least common multiple (LCM) of 3 and 6. After finding these values, we sum them up to get the final answer. Steps:1. Calculate the GCF (or greatest common divisor, GCD) of 3 and 6.2. Calculate the LCM of 3 and 6.3. Sum the GCF and LCM.We can use Python's `math` library, which has functions for GCD and LCM. Let's go ahead and do this using Python code.```pythonimport math# Step 1: Calculate GCF of 3 and 6gcf = math.gcd(3, 6)# Step 2: Calculate LCM of 3 and 6# LCM can be calculated using the formula: LCM(a, b) = abs(a*b) // GCD(a, b)lcm = abs(3 * 6) // gcf# Step 3: Sum the GCF and LCMresult = gcf + lcmprint(result)``````output9```The sum of the greatest common factor of 3 and 6 and the least common multiple of 3 and 6 is (boxed{9}).

answer:To solve for ( y ) given that the vectors (overrightarrow{a} = (4, 2)) and (overrightarrow{b} = (6, y)) are parallel, we need to use the property that parallel vectors are scalar multiples of each other. This means there exists a scalar ( k ) such that:[overrightarrow{b} = k cdot overrightarrow{a}]If we express this in terms of the components, we get:[(6, y) = k cdot (4, 2)]This gives us two equations:[6 = 4k][y = 2k]First, solve for ( k ) from the first equation:[k = frac{6}{4} = frac{3}{2}]Then, substitute ( k ) into the second equation to solve for ( y ):[y = 2 cdot frac{3}{2} = 3]Let's verify this using Python and SymPy:```pythonimport sympy as sp# Define the variablesk, y = sp.symbols('k y')# Define vector a and ba = sp.Matrix([4, 2])b = sp.Matrix([6, y])# Equation for parallel vectorsequation = sp.Eq(b, k * a)# Solve for k and ysolution = sp.solve(equation, (k, y))print(solution)``````output{k: 3/2, y: 3}```The output from the Python code confirms our manual calculation. The value of ( y ) that makes (overrightarrow{a} = (4, 2)) parallel to (overrightarrow{b} = (6, y)) is (boxed{3}).

answer:To solve this problem, we need to follow these steps:1. Solve the quadratic equation (x^2 - 5x + 6 = 0) to find the possible lengths of the third side.2. Verify which of the roots can form a valid triangle with the given sides (3 and 5).3. Compute the perimeter of the triangle by summing up the lengths of its sides.Let's proceed with the steps using Python and SymPy.Step 1: Solve the quadratic equation (x^2 - 5x + 6 = 0).Step 2: Verify the triangle inequality for each root to ensure they form a valid triangle with the sides 3 and 5.Step 3: Compute the perimeter for the valid triangle.Let's implement this in Python.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the quadratic equation x^2 - 5x + 6 = 0quadratic_eq = x2 - 5*x + 6# Solve the quadratic equationroots = sp.solve(quadratic_eq, x)# Given lengths of two sidesside1 = 3side2 = 5# Function to check if three sides form a valid triangledef is_valid_triangle(a, b, c): return a + b > c and a + c > b and b + c > a# Calculate the perimeter for the valid triangleperimeter = Nonefor root in roots: side3 = root.evalf() # Convert to numerical value if is_valid_triangle(side1, side2, side3): perimeter = side1 + side2 + side3 breakprint(perimeter)``````output11.0000000000000```The roots of the equation (x^2 - 5x + 6 = 0) are 2 and 3. Among these, we must check which side length forms a valid triangle with the sides 3 and 5. From the triangle inequality theorem:1. (3 + 5 > 3) (always true)2. (3 + 3 > 5) (true)3. (5 + 3 > 3) (always true)4. (3 + 5 > 2) (always true)5. (3 + 2 > 5) (false)6. (5 + 2 > 3) (true)Thus, only the root 3 forms a valid triangle with the sides 3 and 5.Therefore, the perimeter of the triangle is:[ 3 + 5 + 3 = 11 ]The perimeter of the triangle is (boxed{11}).

answer:To solve this problem, we need to consider the following:1. The cost of buying a bouquet of 7 roses is 20.2. Bill can sell a bouquet of 5 roses for 20.3. We need to calculate the number of bouquets Bill needs to buy in order to make a profit of 1000.Firstly, let's denote the number of bouquets Bill buys as ( n ).Step 1: Calculate the total number of roses Bill buys:Since each bouquet consists of 7 roses, if he buys ( n ) bouquets, the total number of roses he buys is ( 7n ).Step 2: Calculate the total number of bouquets Bill can sell:Since each bouquet he sells consists of 5 roses, the number of bouquets he can sell from the 7 roses per bouquet bought is ( frac{7n}{5} ).Step 3: Calculate the total revenue from selling bouquets:Each bouquet of 5 roses is sold for 20, so his revenue from selling ( frac{7n}{5} ) bouquets will be ( 20 times frac{7n}{5} = 28n ).Step 4: Calculate the total cost of buying bouquets:The cost of buying ( n ) bouquets is ( 20n ).Step 5: Calculate the net profit:The net profit is the total revenue from selling minus the total cost of buying, which is:[ text{Net Profit} = 28n - 20n = 8n ]Step 6: Determine the number of bouquets needed to achieve a net profit of 1000:We need to set up the equation:[ 8n = 1000 ][ n = frac{1000}{8} ]Now let's calculate this using Python to ensure accuracy.```python# Given values required_profit = 1000 profit_per_bouquet = 8 # Calculate the number of bouquets needed number_of_bouquets = required_profit / profit_per_bouquet print(number_of_bouquets)``````output125.0```Bill needs to buy (boxed{125}) bouquets to earn a difference of 1000.

answer:Let's solve this problem step by step using Python and the sympy library. Part 1: Find the Equation of the Line on Which Side BC LiesTo find the equation of the line passing through points B and C, we can use the two-point form of the line equation:[y - y_1 = frac{y_2 - y_1}{x_2 - x_1}(x - x_1)]Given the points B(6, 7) and C(0, 3), we can substitute these coordinates into the formula. Part 2: Calculate the Area of Triangle ABCThe area of a triangle with vertices at ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) can be found using the determinant formula:[text{Area} = frac{1}{2} left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) right|]Given the points A(4, 0), B(6, 7), and C(0, 3), we can substitute these coordinates into the formula.Let's implement this in Python using sympy:```pythonimport sympy as sp# Define variablesx, y = sp.symbols('x y')# Points A, B, and CA = (4, 0)B = (6, 7)C = (0, 3)# Part 1: Equation of the line BC# Using points B(6, 7) and C(0, 3)x1, y1 = Bx2, y2 = C# Calculate the slope (m) and the y-intercept (b) of the lineslope = (y2 - y1) / (x2 - x1)y_intercept = y1 - slope * x1# Equation of the line in the form y = mx + bline_eq = sp.Eq(y, slope * x + y_intercept)# Part 2: Area of triangle ABCx1, y1 = Ax2, y2 = Bx3, y3 = C# Calculate the area using the determinant formulaarea = sp.Rational(1, 2) * abs(x1*(y2 - y3) + x2*(y3 - y1) + x3*(y1 - y2))print("Equation of the line BC:", line_eq)print("Area of triangle ABC:", area)``````outputEquation of the line BC: Eq(y, 0.666666666666667*x + 3.0)Area of triangle ABC: 17```The output provides the solution for both parts of the problem. Part 1: Equation of the Line on Which Side BC LiesThe equation of the line passing through points B(6, 7) and C(0, 3) is:[y = frac{2}{3}x + 3]which can be simplified and written as:[boxed{y = frac{2}{3}x + 3}] Part 2: Area of Triangle ABCThe area of the triangle with vertices A(4, 0), B(6, 7), and C(0, 3) is:[boxed{17}]Thus, the solutions to the problem are ( boxed{y = frac{2}{3}x + 3} ) for the equation of the line BC and ( boxed{17} ) for the area of triangle ABC.

answer:To solve the problem of finding all primes that can be written both as a sum and a difference of two primes, let's break down the solution into structured steps and use Python to ensure accuracy. Steps:1. Generate a List of Prime Numbers: We need a list of prime numbers to work with. We'll generate primes up to a reasonable upper limit.2. Check Sum Condition: We'll identify primes that can be written as the sum of two primes.3. Check Difference Condition: We'll identify primes that can be written as the difference of two primes.4. Find Intersection: The final answer will be the intersection of the primes from the previous two steps. Step 1: Generate a List of Prime NumbersWe'll use the Sieve of Eratosthenes algorithm to generate a list of primes up to some upper limit. Step 2: Check for Sum of Two PrimesFor each prime ( P ), check if ( P = p1 + p2 ) where ( p1 ) and ( p2 ) are primes. Maintain a list for such primes. Step 3: Check for Difference of Two PrimesFor each prime ( P ), check if ( P = |p1 - p2| ) where ( p1 ) and ( p2 ) are primes. Maintain a list for such primes. Step 4: IntersectionFind the common elements from the two lists obtained in steps 2 and 3.Let's start by implementing this step-by-step using Python.```pythondef sieve_of_eratosthenes(limit): """ Sieve of Eratosthenes for generating prime numbers up to a limit. """ is_prime = [True] * (limit + 1) is_prime[0] = is_prime[1] = False for start in range(2, int(limit0.5) + 1): if is_prime[start]: for multiple in range(start*start, limit + 1, start): is_prime[multiple] = False primes = [num for num, prime in enumerate(is_prime) if prime] return primesdef primes_as_sum(primes): """ Find all primes that can be written as the sum of two primes. """ primes_set = set(primes) sum_primes = set() for i in range(len(primes)): for j in range(i, len(primes)): s = primes[i] + primes[j] if s in primes_set: sum_primes.add(s) return sum_primesdef primes_as_difference(primes): """ Find all primes that can be written as the difference of two primes. """ primes_set = set(primes) diff_primes = set() for i in range(len(primes)): for j in range(len(primes)): if i != j: d = abs(primes[i] - primes[j]) if d in primes_set: diff_primes.add(d) return diff_primes# Set an upper limit for generating primesupper_limit = 10000primes = sieve_of_eratosthenes(upper_limit)# Find primes that are sum of two primessum_primes = primes_as_sum(primes)# Find primes that are difference of two primesdiff_primes = primes_as_difference(primes)# Find intersection of the two setsresult_primes = sum_primes.intersection(diff_primes)result_primes_list = sorted(result_primes)print(result_primes_list)``````output[5]```The only prime number below 10,000 that can be written both as a sum of two primes and as a difference of two primes is:(boxed{5})