answer：First, I need to understand the problem. We have a product that is the result of multiplying 1452 by another positive integer ( w ). This product must have ( 2^4 ) and ( 3^3 ) as factors. Additionally, it's given that the smallest possible value of ( w ) is 468. My goal is to find the other factor of the product, which is ( w ).To begin, I'll find the prime factorization of 1452. Breaking it down:[1452 = 2^2 times 3^3 times 13]Next, I'll consider the requirements for the product. It needs to include ( 2^4 ) and ( 3^3 ). Since 1452 already has ( 2^2 ) and ( 3^3 ), the missing factor ( w ) must provide the additional ( 2^2 ) to reach ( 2^4 ).Given that the smallest possible ( w ) is 468, I'll factorize 468:[468 = 2^2 times 3^2 times 13]This confirms that 468 provides the necessary ( 2^2 ) and also includes additional factors of 3 and 13. Therefore, multiplying 1452 by 468 satisfies the requirement for the product to have ( 2^4 ) and ( 3^3 ) as factors.In conclusion, the other factor of the product is 468.

answer：Okay, so I have this problem here where I'm given a function f(x) that's defined on all real numbers, and it's an odd function. I know that when x is greater than 0, f(x) equals x cubed plus x plus 1. The question is asking me what f(x) is when x is less than 0. Hmm, alright, let's break this down.First, I remember that an odd function has a specific property: for every x in its domain, f(-x) is equal to -f(x). That means if I know the value of the function at some positive x, I can find the value at the corresponding negative x by just taking the negative of the positive value. So, if I plug in -x into the function, it should give me the negative of f(x). That seems straightforward.Given that, when x is positive, f(x) is x³ + x + 1. So, if I take a positive number, say 2, f(2) would be 8 + 2 + 1, which is 11. Then, according to the property of odd functions, f(-2) should be -11. So, f(-2) = -f(2) = -11. That makes sense.But wait, the function is defined for all real numbers, so I need a general expression for f(x) when x is negative. Let me think about how to approach this. If x is negative, then -x is positive. Since I know the expression for f when the input is positive, I can use that to find f(-x). Then, using the odd function property, I can relate f(-x) to f(x).So, let's denote x as a negative number. Then, -x is positive, so f(-x) is equal to (-x)³ + (-x) + 1. Let me compute that. (-x)³ is -x³ because when you cube a negative number, it stays negative. Then, (-x) is just -x. So, putting it all together, f(-x) = -x³ - x + 1.But since f is an odd function, f(-x) should also equal -f(x). So, I can set up the equation: -f(x) = -x³ - x + 1. To find f(x), I can multiply both sides of the equation by -1. That gives me f(x) = x³ + x - 1.Wait, let me double-check that. If I have -f(x) = -x³ - x + 1, then multiplying both sides by -1 gives f(x) = x³ + x - 1. Yeah, that seems right. So, for negative x, f(x) is x³ + x - 1.Let me test this with an example to make sure. Let's take x = -1. Then, f(-1) should be (-1)³ + (-1) - 1, which is -1 -1 -1 = -3. Now, using the odd function property, f(-1) should be equal to -f(1). Let's compute f(1): f(1) = 1³ + 1 + 1 = 1 + 1 + 1 = 3. So, -f(1) is -3, which matches f(-1) = -3. That checks out.Another example: x = -2. f(-2) should be (-2)³ + (-2) - 1 = -8 -2 -1 = -11. On the other hand, f(2) is 8 + 2 + 1 = 11, so -f(2) is -11, which again matches. So, it seems like my expression for f(x) when x is negative is correct.I think I've got it. So, when x is less than 0, f(x) is x³ + x - 1. That makes sense because the odd function property requires that the function is symmetric about the origin, so the negative input flips the sign of the output, but also accounts for the specific form of the function when x is positive.Just to recap, the steps I took were:1. Recognize that f is an odd function, so f(-x) = -f(x).2. For x < 0, let’s denote x as negative, so -x is positive.3. Use the given expression for f when the input is positive: f(-x) = (-x)³ + (-x) + 1.4. Simplify f(-x) to get -x³ - x + 1.5. Set this equal to -f(x) because of the odd function property.6. Solve for f(x) by multiplying both sides by -1, resulting in f(x) = x³ + x - 1.7. Verify with specific examples to ensure correctness.This process seems solid. I don't see any mistakes in the logic or calculations. The key was understanding how the odd function property ties together the function's values at positive and negative inputs. It's crucial to remember that for odd functions, the function's graph is symmetric about the origin, which means if you rotate the graph 180 degrees around the origin, it looks the same. This symmetry is reflected in the algebraic relationship f(-x) = -f(x).Another thing to note is that when dealing with odd functions, certain terms in the function will behave differently when the input is negated. For example, in the given function f(x) = x³ + x + 1 for x > 0, the x³ and x terms are odd-powered, so they will change sign when x is replaced with -x. However, the constant term, which is 1, doesn't have an x attached, so it remains the same. But because of the odd function property, we have to adjust for that constant term when finding f(x) for negative x.Wait a second, let me think about that constant term. In the original function for x > 0, there's a +1. When I plug in -x into the function, I get -x³ - x + 1. But since f is odd, f(-x) should be -f(x). So, if f(-x) is -x³ - x + 1, then -f(x) must equal that. Therefore, f(x) is x³ + x - 1. So, the constant term flips sign because of the negative sign in front of f(x). That makes sense because the constant term isn't tied to x, so it doesn't change sign on its own; instead, the negative sign in front of f(x) flips its contribution.This is an important point because sometimes when dealing with odd functions, especially polynomials, the constant term can be tricky. If the function had only odd-powered terms, like f(x) = x³ + x, then it would naturally be an odd function without any adjustments. However, the presence of the constant term complicates things because constants aren't odd or even in the same way. So, in this case, to maintain the odd function property, the constant term must be negated when moving from positive to negative x.Let me also consider the general form of an odd function. An odd function can be expressed as a sum of terms where each term is an odd power of x multiplied by a coefficient. So, for example, f(x) = a x³ + b x + c x⁻¹ + ... are all odd functions. However, in this case, the given function for x > 0 is f(x) = x³ + x + 1, which includes a constant term. Since the constant term isn't an odd-powered term, it doesn't naturally satisfy the odd function property. Therefore, to make the entire function odd, we have to adjust the constant term when defining f(x) for negative x.This adjustment is exactly what we did when we found that f(x) = x³ + x - 1 for x < 0. The constant term went from +1 to -1, ensuring that the function remains odd. If we hadn't adjusted the constant term, the function wouldn't satisfy f(-x) = -f(x) because the constant term would remain the same, breaking the symmetry required for an odd function.So, in summary, when dealing with an odd function that has a constant term, we have to be careful to adjust that term appropriately when defining the function for negative inputs. The constant term doesn't change sign on its own, so we have to account for it by negating it when applying the odd function property.Another aspect to consider is the continuity of the function at x = 0. Since f(x) is defined on all real numbers, it must be continuous at x = 0. Let's check that. For an odd function, f(0) must be 0 because f(-0) = -f(0) implies f(0) = -f(0), which can only be true if f(0) = 0.Looking at the given function for x > 0, f(x) = x³ + x + 1. If we plug in x = 0, we get f(0) = 0 + 0 + 1 = 1. But wait, that's a problem because for an odd function, f(0) should be 0. This seems contradictory. Did I make a mistake somewhere?Hold on, the problem states that f(x) is an odd function defined on ℝ, and when x > 0, f(x) = x³ + x + 1. It doesn't specify what f(0) is, but since it's an odd function, f(0) must be 0. However, according to the given expression for x > 0, f(0) would be 1. That seems inconsistent.Is there a mistake in the problem statement? Or perhaps, is the function only defined as x³ + x + 1 for x > 0, and we're supposed to define it for x < 0 such that it's odd, regardless of what happens at x = 0? Because if f(0) is supposed to be 0, but according to the given expression, it's 1, that's a conflict.Wait, maybe the function isn't defined as x³ + x + 1 for x ≥ 0, but only for x > 0. So, at x = 0, it's a separate case. Since f is odd, f(0) must be 0, so we can define f(0) = 0 separately. That way, the function is consistent. So, for x > 0, f(x) = x³ + x + 1, for x = 0, f(x) = 0, and for x < 0, f(x) = x³ + x - 1. That makes sense because at x = 0, the function is 0, satisfying the odd function property.So, in this case, the function isn't x³ + x + 1 for all x ≥ 0, but only for x > 0. At x = 0, it's defined separately as 0 to maintain the odd function property. That resolves the inconsistency I noticed earlier.Therefore, when x < 0, f(x) = x³ + x - 1, and at x = 0, f(x) = 0. This ensures that the function is odd everywhere on ℝ.Let me just confirm this with another example. Take x = 0.5, which is positive. f(0.5) = (0.5)³ + 0.5 + 1 = 0.125 + 0.5 + 1 = 1.625. Then, f(-0.5) should be -1.625. According to our expression for x < 0, f(-0.5) = (-0.5)³ + (-0.5) - 1 = -0.125 - 0.5 - 1 = -1.625. Perfect, that matches.Another test: x = -3. f(-3) should be (-3)³ + (-3) - 1 = -27 - 3 - 1 = -31. And f(3) is 27 + 3 + 1 = 31, so -f(3) is -31, which matches f(-3). Everything seems consistent.I think I've covered all the bases here. I started by recalling the definition of an odd function, applied it to the given function for positive x, derived the expression for negative x, checked for consistency with specific examples, and even considered the behavior at x = 0 to ensure the function is properly defined and continuous there. It's a good practice to verify each step, especially when dealing with function properties like oddness or evenness, as small oversights can lead to incorrect conclusions.In conclusion, the key takeaway is that for an odd function, knowing the expression for positive x allows us to determine the expression for negative x by applying the property f(-x) = -f(x). Additionally, ensuring that the function is properly defined at x = 0 is crucial for maintaining the odd function property and overall continuity.

answer：Okay, so I have this problem here about negating a proposition. The original statement is "There exists an ( x in mathbb{R} ), such that ( e^x < x )." I need to find its negation from the given options: A, B, C, D. Let me think through this step by step.First, I remember that in logic, the negation of an existential quantifier ("there exists") is a universal quantifier ("for all"). So, if the original statement is saying that there is at least one real number ( x ) where ( e^x ) is less than ( x ), then the negation would be that for every real number ( x ), ( e^x ) is not less than ( x ). In other words, if the original is ( exists x in mathbb{R}, e^x < x ), then the negation should be ( forall x in mathbb{R}, e^x geq x ). That makes sense because if it's not true that there's some ( x ) where ( e^x < x ), then for all ( x ), ( e^x ) must be greater than or equal to ( x ).Looking at the options, option B says exactly that: "For all ( x in mathbb{R} ), ( e^x geq x )." So, that should be the correct negation.Let me double-check to make sure I didn't make a mistake. Sometimes, when dealing with inequalities, it's easy to flip the wrong part. The original statement is ( e^x < x ). The negation of ( e^x < x ) is ( e^x geq x ). So, yes, that part is correct.Also, just to be thorough, let me consider the other options:- Option A: "There exists an ( x in mathbb{R} ), such that ( e^x > x )." This is not the negation because the negation should cover all ( x ), not just the existence of one. - Option C: "There exists an ( x in mathbb{R} ), ( e^x geq x )." Again, this is about existence, not universality, so it's not the negation. - Option D: "For all ( x in mathbb{R} ), ( e^x > x )." This is close, but it's not exactly the negation. The negation of ( e^x < x ) is ( e^x geq x ), which includes both ( e^x > x ) and ( e^x = x ). So, D is too strong because it doesn't allow for equality.Therefore, after considering all options, B is indeed the correct negation of the original proposition.Just to add another layer of verification, maybe I can think about the graphs of ( e^x ) and ( x ). The exponential function ( e^x ) grows much faster than the linear function ( x ). For negative values of ( x ), ( e^x ) approaches zero, which is greater than ( x ) since ( x ) is negative. For positive ( x ), ( e^x ) grows much faster than ( x ), so ( e^x ) will always be above ( x ). At ( x = 0 ), ( e^0 = 1 ), which is greater than 0. So, in fact, ( e^x ) is always greater than or equal to ( x ) for all real numbers ( x ). That means the original statement is false because there does not exist an ( x ) where ( e^x < x ). Therefore, the negation, which is option B, is actually a true statement. This further confirms that my reasoning was correct. The negation flips the existential quantifier to a universal one and negates the inequality, resulting in option B being the correct choice.I think I've covered all the angles here. I've considered the logical structure, checked each option, verified with an example, and even thought about the graphical interpretation. I'm pretty confident that B is the right answer.**Final Answer**boxed{B}

answer：Okay, so I need to figure out the smaller angle between the hour and minute hands of a clock at 7:15 p.m. Hmm, let's break this down step by step. I remember that clocks are circles, and circles have 360 degrees. There are 12 hours on a clock, so each hour mark must represent 360 divided by 12, which is 30 degrees. That means every hour, the hour hand moves 30 degrees. Now, at 7:15 p.m., the minute hand is pointing at the 3, right? Because 15 minutes is a quarter of an hour, so it's at the 3. That should be 90 degrees from the top (12 o'clock position) because 15 minutes is a quarter of 60, and a quarter of 360 is 90. So, the minute hand is at 90 degrees.But what about the hour hand? At exactly 7:00 p.m., the hour hand is at 7, which is 7 times 30 degrees, so that's 210 degrees. But it's not exactly 7:00; it's 7:15. So, the hour hand has moved a bit past the 7. How much has it moved? Well, in 60 minutes, the hour hand moves 30 degrees. So, in one minute, it moves 30 divided by 60, which is 0.5 degrees per minute. Since it's 15 minutes past 7, the hour hand has moved 15 times 0.5 degrees. Let me calculate that: 15 times 0.5 is 7.5 degrees. So, the hour hand is at 210 degrees plus 7.5 degrees, which is 217.5 degrees.Now, to find the angle between them, I subtract the smaller angle from the larger one. The minute hand is at 90 degrees, and the hour hand is at 217.5 degrees. So, 217.5 minus 90 is 127.5 degrees. Wait, but I need to make sure this is the smaller angle. Since a full circle is 360 degrees, the larger angle would be 360 minus 127.5, which is 232.5 degrees. Clearly, 127.5 is smaller than 232.5, so 127.5 degrees is the smaller angle between the hour and minute hands at 7:15 p.m.I think that makes sense. Let me double-check my calculations. Each hour is 30 degrees, so 7 hours is 210 degrees. 15 minutes is a quarter of an hour, so the hour hand moves a quarter of 30 degrees, which is 7.5 degrees. Adding that to 210 gives 217.5 degrees. The minute hand at 15 minutes is 90 degrees. The difference is 127.5 degrees, which is less than 180, so it's indeed the smaller angle. Yeah, that seems right.

answer：First, I need to determine the total rent paid by all four friends before any changes. Since the average rent is 800, the total rent is 4 multiplied by 800, which equals 3200.Next, after one person's rent is increased by 25%, the new average rent becomes 850. Therefore, the new total rent is 4 multiplied by 850, which equals 3400.The difference between the new total rent and the original total rent is 3400 minus 3200, resulting in a 200 increase.This 200 increase represents a 25% increase on the original rent of the friend whose rent was raised. To find the original rent, I can set up the equation 0.25R = 200, where R is the original rent.Solving for R, I divide both sides of the equation by 0.25, which gives R = 200 / 0.25 = 800.Therefore, the original rent of the friend whose rent was increased is 800.

answer：Okay, so I need to figure out if the statement "If a is greater than b, then a times c squared is greater than b times c squared" is true or false. Hmm, let's break this down step by step.First, I know that a and b are real numbers, and c is also a real number because it's squared. The statement is saying that if a is bigger than b, then when you multiply both a and b by c squared, the inequality still holds. That makes sense intuitively because if you have two numbers and you multiply both by the same positive number, the inequality should stay the same. But wait, c squared is always positive, right? Because any real number squared is non-negative, and it's only zero if c is zero.So, if c is not zero, then c squared is positive. In that case, multiplying both sides of the inequality a > b by c squared should preserve the inequality, so ac² > bc². That part seems straightforward.But what if c is zero? If c is zero, then c squared is zero. So, multiplying both a and b by zero would give us 0 > 0, which is not true because 0 is equal to 0. So, in this specific case, the conclusion doesn't hold. That means the original statement isn't always true because there's a scenario where it fails.Wait, but does the original statement specify any conditions on c? It just says a, b are real numbers, but it doesn't say anything about c. So, c could be zero, which would make the conclusion false even though the premise a > b is true. That means the implication isn't always true.I think I need to consider the definition of an implication in logic. An implication "If P, then Q" is only false when P is true and Q is false. In this case, P is "a > b" and Q is "ac² > bc²". If there's any situation where a > b is true but ac² > bc² is false, then the implication is false.As I thought earlier, if c is zero, then ac² = bc² = 0, so ac² is not greater than bc². Therefore, when c is zero, P is true (assuming a > b) and Q is false. This makes the implication false in this case.But wait, maybe I'm missing something. Is c squared always non-negative? Yes, because squaring any real number gives a non-negative result. So, c squared is either positive or zero. If c squared is positive, then multiplying both sides of a > b by c squared preserves the inequality. But if c squared is zero, then both sides become zero, and the inequality doesn't hold.So, the statement "If a > b, then ac² > bc²" is only true when c is not zero. But since the statement doesn't specify that c is not zero, we have to consider all possible real numbers c, including zero. Therefore, the statement isn't universally true because there's a case where it fails.I think I need to check if there's any other way to interpret the statement. Maybe c is supposed to be a non-zero real number? But the problem statement just says a, b are real numbers, and it doesn't specify c. So, I have to assume c can be any real number, including zero.Another angle: maybe the statement is considered true if it's true for all cases except when c is zero. But in logic, an implication is only true if it holds for all possible cases. If there's even one case where it fails, the implication is false.So, summarizing my thoughts: The statement is true when c is not zero because multiplying both sides of an inequality by a positive number preserves the inequality. However, when c is zero, multiplying both sides by zero results in both sides being equal, which makes the conclusion false. Since there exists a case where the premise is true but the conclusion is false, the entire implication is false.I think that's the correct reasoning. The statement isn't always true because it depends on the value of c. If c is zero, it fails. Therefore, the proposition is false.