answer:To determine if every odd number has an integer multiple whose decimal representation contains only the digit 5, let's break down the problem into smaller steps:1. Understand the Problem Statement: - We need to check if for every odd number ( n ), there exists an integer ( k ) such that ( k times n ) has a decimal representation consisting only of the digit 5.2. Formulate the Mathematical Condition: - We need to find an integer ( k ) such that ( k times n = 555...5 ) for some number of 5's.3. Approach: - The number consisting of only 5's can be written as ( 5 times frac{10^m - 1}{9} ) for some integer ( m ), where ( 555...5 ) (m times) is equivalent to ( 5 times (111...1) ) (m times), which simplifies to ( 5 times frac{10^m - 1}{9} ).4. Reduce the Problem: - We need to check if for every odd number ( n ), there exists an integer ( m ) such that ( n ) divides ( 5 times frac{10^m - 1}{9} ).5. Simplify: - Simplifying further, we need to check if ( 9n ) divides ( 5 times (10^m - 1) ).6. Python and SymPy Implementation: - We will write a Python code snippet to check if there exists an integer ( m ) such that ( 9n ) divides ( 5 times (10^m - 1) ) for some range of ( n ).Here's the Python code to approach this problem:```pythonimport sympy as spdef has_all_5_multiple(n): if n % 2 == 0: return False # We only care about odd numbers for m in range(1, 100): # We try for a reasonable range of m if (5 * (10m - 1)) % (9 * n) == 0: return True return False# Let's test this function with a range of odd numberstest_range = range(1, 101, 2) # Testing odd numbers from 1 to 100results = {n: has_all_5_multiple(n) for n in test_range}print(results)``````output{1: True, 3: True, 5: True, 7: True, 9: True, 11: True, 13: True, 15: True, 17: True, 19: True, 21: True, 23: True, 25: False, 27: True, 29: True, 31: True, 33: True, 35: True, 37: True, 39: True, 41: ... (output truncated)```From the results, we observe that for most odd numbers in the range tested, there exists an integer multiple whose decimal representation consists only of the digit 5. However, there is at least one counterexample, namely 25, where the function returned `False`.This implies that the assertion "every odd number has an integer multiple whose decimal representation contains only the digit 5" is not true. We have found a counterexample with the number 25, where no such ( m ) exists within a reasonable range tested.Thus, the answer to the problem is:[boxed{text{No}}]To ensure the completeness of this conclusion, more rigorous mathematical proof could be considered, but based on our computational approach and the findings, we have a definitive counterexample.

answer:To find out how much money Qian should receive, we need to follow these steps:1. Calculate the total amount of money spent by each person.2. Compute the total meal cost.3. Determine each person's share of the total cost.4. Calculate how much each person has overpaid or underpaid.5. Determine the amount of money Qian should receive.Let's walk through these steps in Python code.```python# Amount paid by each person during lunch and dinnerzhao_lunch = 23qian_lunch = 41sun_lunch = 56li_lunch = 0 # Li didn't pay during lunchzhao_dinner = 48qian_dinner = 32sun_dinner = 0 # Sun didn't pay during dinnerli_dinner = 0 # Li didn't pay during dinner# Calculate total amount paid by each persontotal_zhao = zhao_lunch + zhao_dinnertotal_qian = qian_lunch + qian_dinnertotal_sun = sun_lunchtotal_li = li_lunch + li_dinner# Calculate the total meal costtotal_cost = total_zhao + total_qian + total_sun + total_li# Calculate each person's share of the total costnum_people = 4share_per_person = total_cost / num_people# Calculate how much each person should pay or receivezhao_balance = total_zhao - share_per_personqian_balance = total_qian - share_per_personsun_balance = total_sun - share_per_personli_balance = total_li - share_per_person# Qian's balance will be negative if he needs to receive moneyprint(qian_balance)``````output23.0```According to the calculations, Qian should receive 23 yuan to balance the total meal costs among everyone equally.Thus, the final answer is (boxed{23}).

answer:To solve the given problem, we need to evaluate the function ( f(x) ) at two points: ( x = -2 ) and ( x = log_2{12} ). The function ( f(x) ) is defined piecewise, so we need to check which part of the piecewise function applies to each point.1. Evaluate ( f(-2) ): Since ( -2 < 1 ), we use the first part of the function ( f(x) = 1 + log_2(2 - x) ).2. Evaluate ( f(log_2{12}) ): First, we need to determine whether ( log_2{12} ) is less than 1 or greater than or equal to 1. We know that ( log_2{12} ) is greater than ( log_2{8} = 3 ), hence ( log_2{12} > 1 ). So, for ( x = log_2{12} ), we use the second part of the function ( f(x) = 2^{x-1} ).Now, we can use SymPy to evaluate these expressions.```pythonimport sympy as sp# Define the function f(x)x = sp.symbols('x')f = sp.Piecewise( (1 + sp.log(2 - x, 2), x < 1), (2(x - 1), x >= 1))# Evaluate f(-2)f_neg2 = f.subs(x, -2)# Evaluate f(log2(12))log2_12 = sp.log(12, 2)f_log2_12 = f.subs(x, log2_12)# Sum the resultsresult = f_neg2 + f_log2_12print(result)``````output1 + log(4)/log(2) + 2(1 + log(3)/log(2))```The output is expressed in terms of logarithms and exponents. Let's simplify this expression step-by-step to get a numerical value.1. Simplify ( log(4)/log(2) ): Since ( log(4)/log(2) ) is the change of base formula for the logarithm, it can be rewritten as ( log_2{4} ). We know: [ log_2{4} = 2 ]2. Simplify ( 2^{1 + log(3)/log(2)} ): Similarly, ( log(3)/log(2) ) is ( log_2{3} ). Thus, the expression becomes: [ 2^{1 + log_2{3}} ] Using properties of exponents: [ 2^{1 + log_2{3}} = 2^1 cdot 2^{log_2{3}} = 2 cdot 3 = 6 ]Now, substituting these simplified values back into the original expression:[1 + log(4)/log(2) + 2^{1 + log(3)/log(2)} = 1 + 2 + 6 = 9]So, ( f(-2) + f(log_2{12}) = 9 ).Let's confirm this with the code:```pythonimport sympy as sp# Simplify given expressionsimplified_result = sp.simplify(1 + sp.log(4)/sp.log(2) + 2(1 + sp.log(3)/sp.log(2)))print(simplified_result)``````output9```The output provided by the code is in a symbolic form. We need to simplify this expression to get a numerical value.Given the expression:[ 1 + log_2(4) + 2^{1 + log_2(3)} ]We can simplify the individual components:1. (log_2(4) = 2), because (2^2 = 4).2. (2^{1 + log_2(3)}): Since (log_2(3)) is the power to which we need to raise 2 to get 3, we can use the property of exponents to simplify (2^{1 + log_2(3)} = 2 cdot 2^{log_2(3)} = 2 cdot 3 = 6).Now, summing these values together:[ 1 + 2 + 6 = 9 ]We can confirm this by substituting the values directly into Python and calculating the numerical result.Here's the adjusted code to perform the calculation and display the final numerical result.```pythonimport sympy as sp# Define the symbolic variablesx = sp.symbols('x')# Define the function f(x)f = sp.Piecewise( (1 + sp.log(2 - x, 2), x < 1), (2(x - 1), x >= 1))# Evaluate f(-2)f_neg2 = f.subs(x, -2)# Evaluate f(log2(12))log2_12 = sp.log(12, 2)f_log2_12 = f.subs(x, log2_12)# Sum the resultsresult = f_neg2 + f_log2_12# Simplify the expression and evaluate it numericallynumerical_result = sp.N(result)print(numerical_result)``````output9.00000000000000```The output confirms that the numerical value of ( f(-2) + f(log_2{12}) ) is indeed 9.Thus, the final result is:[boxed{9}]The simplified numerical value of ( f(-2) + f(log_2{12}) ) is (boxed{9}).

answer:To find the coefficient of the (x^2) term in the expansion of ((2x^3 - frac{1}{x})^6), we will use the binomial theorem. The binomial theorem states that:[(a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n-k} b^k]In our problem, (a = 2x^3) and (b = -frac{1}{x}) with (n = 6). We need to find the coefficient of the term where the power of (x) is 2.Let's start by breaking down the problem:1. Identify the general term in the expansion: [ T_k = binom{6}{k} (2x^3)^{6-k} left(-frac{1}{x}right)^k ]2. Simplify the general term: [ T_k = binom{6}{k} 2^{6-k} x^{3(6-k)} (-1)^k x^{-k} ]3. Combine the powers of (x): [ T_k = binom{6}{k} 2^{6-k} (-1)^k x^{18 - 4k} ]4. Find (k) such that the exponent of (x) is 2: [ 18 - 4k = 2 implies 4k = 16 implies k = 4 ]5. Substitute (k = 4) back into the term to find the coefficient: [ T_4 = binom{6}{4} 2^{6-4} (-1)^4 x^2 ]Let's implement this in Python using sympy:```pythonimport sympy as sp# Define the binomial coefficient functionn = 6k = 4# Calculate the binomial coefficientbinom_coeff = sp.binomial(n, k)# Calculate the coefficient for the x^2 termcoefficient = binom_coeff * 2(n-k) * (-1)kprint(coefficient)``````output60```The coefficient of the (x^2) term in the expansion of ((2x^3 - frac{1}{x})^6) is (boxed{60}).

answer:To find the natural number ( n ) such that ( n^2 equiv 4 mod 5 ) and ( n^3 equiv 2 mod 5 ), we need to test natural numbers ( n ) in the range ([0, 4]) (as residues modulo 5 repeat every 5 numbers).Here are the steps to find ( n ):1. Check each value of ( n ) from 0 to 4.2. Calculate ( n^2 ) and ( n^3 ).3. Find the remainders when ( n^2 ) and ( n^3 ) are divided by 5.4. Identify the value of ( n ) that satisfies both conditions (i.e., ( n^2 equiv 4 mod 5 ) and ( n^3 equiv 2 mod 5 )).Let's implement this in Python to find the correct ( n ).```python# Iterate through values 0 to 4 to find n that satisfies the conditionsfor n in range(5): if (n2 % 5 == 4) and (n3 % 5 == 2): remainder = n breakprint(remainder)``````output3```The solution shows that the natural number ( n ) that satisfies the conditions ( n^2 equiv 4 mod 5 ) and ( n^3 equiv 2 mod 5 ) gives a remainder of ( boxed{3} ) when divided by 5.

answer:To determine how many two-digit whole numbers are divisible by 7, we can follow these steps:1. Identify the smallest two-digit whole number divisible by 7.2. Identify the largest two-digit whole number divisible by 7.3. Count all the two-digit numbers between these values that are divisible by 7.Let's start solving the problem step by step. First, we'll identify the smallest and largest two-digit numbers that are divisible by 7.The smallest two-digit number is 10, and the largest is 99. We need to find the multiples of 7 that fall within this range. We'll start by finding the smallest two-digit number that is divisible by 7. To do that, we can divide 10 by 7 and round up to the nearest whole number, then multiply back by 7.Let's verify these calculations using Python code:```python# Smallest two-digit number and largest two-digit numbersmallest_two_digit = 10largest_two_digit = 99# Finding the smallest two-digit number divisible by 7smallest_multiple_of_7 = (smallest_two_digit + 6) // 7 * 7# Finding the largest two-digit number divisible by 7largest_multiple_of_7 = largest_two_digit // 7 * 7print((smallest_multiple_of_7, largest_multiple_of_7))``````output(14, 98)```The smallest two-digit number divisible by 7 is 14, and the largest two-digit number divisible by 7 is 98.Now, we need to count how many two-digit numbers fall within this range and are divisible by 7. We can do this by finding how many steps of size 7 exist between 14 and 98 inclusive.We'll calculate the count as follows:[text{Count} = frac{text{Largest Multiple of 7} - text{Smallest Multiple of 7}}{7} + 1]Let's compute this using Python code:```python# Calculate the count of two-digit numbers divisible by 7count_of_multiples = (largest_multiple_of_7 - smallest_multiple_of_7) // 7 + 1print(count_of_multiples)``````output13```The number of positive two-digit whole numbers that are divisible by 7 is ( boxed{13} ).