answer：Okay, so I have this problem where there's a number 12 on the computer screen, and every second, I can either multiply or divide it by 2 or 3. After exactly one minute, which is 60 seconds, I need to figure out if the number on the screen could be 54.First, let me understand the operations. Every second, I can choose to multiply or divide by 2 or 3. So, each operation changes the number by a factor of 2 or 3, either increasing or decreasing it. That means every second, the number can either become twice as big, half as big, three times as big, or a third of its current value.Starting with 12, I need to see if after 60 such operations, I can end up with 54. Hmm, okay. Let me break this down.Maybe I should look at the prime factors of the numbers involved. 12 is 2^2 * 3^1, and 54 is 2^1 * 3^3. So, both numbers have the same prime factors, just different exponents.Now, each operation either multiplies or divides by 2 or 3. That means each operation affects the exponents of 2 or 3 in the prime factorization. For example, multiplying by 2 increases the exponent of 2 by 1, dividing by 2 decreases it by 1, and similarly for 3.So, starting from 12, which is 2^2 * 3^1, each operation will change either the exponent of 2 or the exponent of 3 by +1 or -1.After 60 operations, I need to get to 54, which is 2^1 * 3^3. So, I need to change the exponents from (2,1) to (1,3). That means decreasing the exponent of 2 by 1 and increasing the exponent of 3 by 2.But wait, each operation only changes one exponent at a time. So, to decrease the exponent of 2 by 1, I need one division by 2. To increase the exponent of 3 by 2, I need two multiplications by 3. So, that's a total of three operations: one division by 2 and two multiplications by 3.But hold on, the problem says that exactly one minute later, which is 60 seconds, the number appears. So, I have to perform 60 operations, not just three. That complicates things because I have to somehow perform 60 operations that result in a net change of decreasing the exponent of 2 by 1 and increasing the exponent of 3 by 2.Hmm, so the total number of operations is 60, but the net effect is only a small change in exponents. That suggests that many of the operations might cancel each other out. For example, I could multiply by 2 and then divide by 2, which would cancel out. Similarly for 3.So, perhaps I need to think about the parity of the number of operations. Each operation changes the exponent of either 2 or 3 by 1. So, each operation changes the total number of exponents (sum of exponents) by 1. Starting from 12, which has exponents summing to 2 + 1 = 3, after 60 operations, the sum of exponents would be 3 + 60k, where k is some integer. But since each operation changes the sum by +1 or -1, the total change after 60 operations would be 60 steps of +1 or -1.Wait, but 60 is an even number. So, the total change in the sum of exponents would be even. Starting from 3, which is odd, adding or subtracting an even number would result in an odd number. But 54 has exponents summing to 1 + 3 = 4, which is even. So, that's a problem because we started with an odd sum and after 60 operations, we should still have an odd sum, but 54 has an even sum.Therefore, it's impossible to reach 54 from 12 in 60 operations because the parity of the sum of exponents doesn't match.Wait, let me double-check that. The initial sum is 3 (odd). Each operation changes the sum by +1 or -1, so after an even number of operations (60), the total change would be even. Therefore, the final sum would be 3 + even = odd. But 54 has a sum of 4, which is even. So, yes, it's impossible.But maybe I'm missing something. What if some operations cancel out? For example, multiplying by 2 and then dividing by 2 would cancel out, but that would still count as two operations, keeping the sum the same. So, even if I do such pairs, the total number of operations is still 60, and the sum would still have to be odd.Therefore, I think the conclusion is that it's impossible to reach 54 from 12 in exactly 60 operations because the parity of the sum of exponents doesn't match.

answer：First, I recognize that 1.6 is equivalent to 16 divided by 10. This means that 1.6 × 213.0 can be rewritten as (16 × 213.0) divided by 10.Given that 213 × 16 equals 3408, I can substitute this value into the equation. So, (16 × 213.0) is 3408.Finally, I divide 3408 by 10 to find the result of 1.6 × 213.0, which is 340.8.

answer：Alright, so I have this geometry problem here, and I need to figure out which of the given propositions are correct. Let me try to understand the setup first.We have two planes, α and β, that are perpendicular to each other, and their intersection is the line l. There's a point P that lies on plane α but is not on the line l. So, P is somewhere in plane α, but not where α and β meet.Now, there are four propositions given, numbered ① to ④, and I need to determine which ones are correct. Let me go through each one step by step.Starting with proposition ①: "The plane passing through P and perpendicular to l is perpendicular to β."Hmm, okay. So, if I imagine a plane that goes through point P and is perpendicular to line l, which is the intersection of α and β. Since α and β are already perpendicular, does that mean any plane perpendicular to l would also be perpendicular to β?Wait, I think there's a theorem about this. If two planes are perpendicular, then a plane that's perpendicular to their line of intersection should be perpendicular to both planes. So, if I have a plane that's perpendicular to l, it should be perpendicular to both α and β. But in this case, the plane is passing through P, which is in α but not on l. So, does that affect anything?I think it still holds because the plane is constructed to be perpendicular to l, regardless of where P is. So, this plane should indeed be perpendicular to β. So, proposition ① seems correct.Moving on to proposition ②: "The line passing through P and perpendicular to l is perpendicular to β."Okay, so now instead of a plane, we're talking about a line that goes through P and is perpendicular to l. Is this line necessarily perpendicular to β?Well, let's think about the properties of perpendicular lines and planes. If a line is perpendicular to another line, does that make it perpendicular to the entire plane? Not necessarily. It just means it's perpendicular to that specific line.But in this case, β is a plane, and l is the intersection of α and β. So, if a line is perpendicular to l, does that mean it's perpendicular to the entire plane β?I don't think so. Because β has infinitely many lines, not just l. For a line to be perpendicular to β, it needs to be perpendicular to every line in β, which is a much stronger condition. So, just being perpendicular to l doesn't guarantee that.Therefore, proposition ② is probably incorrect.Next, proposition ③: "The line passing through P and perpendicular to α is parallel to β."Alright, so we have a line that goes through P and is perpendicular to α. Since α and β are perpendicular, what does that mean for this line?If a line is perpendicular to α, it must be parallel to the normal vector of α. But since α and β are perpendicular, their normal vectors are also perpendicular. So, does that mean the line is parallel to β?Wait, if a line is perpendicular to α, it's parallel to the normal vector of α. Since β is perpendicular to α, the normal vector of α lies within β. Therefore, the line, being parallel to the normal vector of α, would be parallel to β.Yes, that makes sense. So, proposition ③ is correct.Finally, proposition ④: "The line passing through P and perpendicular to β is within α."So, we have a line that goes through P and is perpendicular to β. Since P is in α, and α is perpendicular to β, does this line lie entirely within α?Well, if a line is perpendicular to β, it must be parallel to the normal vector of β. But since α and β are perpendicular, the normal vector of β lies within α. Therefore, any line perpendicular to β starting from a point in α should lie within α.Yes, that seems right. So, proposition ④ is correct.Let me just recap:- Proposition ①: Correct.- Proposition ②: Incorrect.- Proposition ③: Correct.- Proposition ④: Correct.So, the correct propositions are ①, ③, and ④.**Final Answer**The correct propositions are boxed{①③④}.

answer：First, I need to understand the universal set ( U ). It is defined as all positive integers less than 6, so ( U = {1, 2, 3, 4, 5} ).Next, I have sets ( A = {1, 3} ) and ( B = {3, 5} ). I need to find the union of ( A ) and ( B ), which includes all elements that are in either ( A ) or ( B ). Therefore, ( A cup B = {1, 3, 5} ).Now, to find the complement of ( A cup B ) in ( U ), I need to identify all elements in ( U ) that are not in ( A cup B ). The elements in ( U ) are ( 1, 2, 3, 4, 5 ), and the elements in ( A cup B ) are ( 1, 3, 5 ). The elements not in ( A cup B ) are ( 2 ) and ( 4 ).Therefore, the complement of ( A cup B ) in ( U ) is ( {2, 4} ).

answer：Alright, so I have this problem here about vectors. Let me try to figure it out step by step. Okay, the vectors given are a, b, and c. Vector a is (√3, 1), vector b is (0, 1), and vector c is (k, √3). The condition is that a + 2b is perpendicular to c. I need to find the value of k.First, I remember that when two vectors are perpendicular, their dot product is zero. So, if I can find the vectors a + 2b and c, then take their dot product and set it equal to zero, I should be able to solve for k.Let me write down what a + 2b is. Vector a is (√3, 1), and vector b is (0, 1). So, 2b would be 2 times (0, 1), which is (0*2, 1*2) = (0, 2). Now, adding a and 2b together: a + 2b = (√3 + 0, 1 + 2) = (√3, 3). Okay, so a + 2b is the vector (√3, 3).Now, vector c is given as (k, √3). So, I need to take the dot product of (√3, 3) and (k, √3) and set it equal to zero because they are perpendicular.The dot product is calculated by multiplying the corresponding components and then adding them together. So, that would be (√3 * k) + (3 * √3). Let me write that out: √3 * k + 3 * √3 = 0.Hmm, so I have √3 * k + 3√3 = 0. I can factor out √3 from both terms, which gives me √3(k + 3) = 0. Since √3 is not zero, the equation simplifies to k + 3 = 0. Therefore, solving for k, I subtract 3 from both sides and get k = -3.Wait, let me double-check that. If I plug k = -3 back into vector c, it becomes (-3, √3). Then, the dot product with a + 2b, which is (√3, 3), should be zero. Calculating that: √3*(-3) + 3*√3 = -3√3 + 3√3 = 0. Yep, that checks out.So, I think I did that correctly. I found a + 2b, took the dot product with c, set it equal to zero, solved for k, and verified the solution. It seems solid. I don't think I made any mistakes here. Maybe I should just recap the steps to make sure.1. Calculate a + 2b: - a = (√3, 1) - 2b = (0, 2) - a + 2b = (√3 + 0, 1 + 2) = (√3, 3)2. Vector c = (k, √3)3. Dot product of (a + 2b) and c: - (√3)(k) + (3)(√3) = √3k + 3√34. Set dot product equal to zero: - √3k + 3√3 = 05. Factor out √3: - √3(k + 3) = 06. Since √3 ≠ 0, k + 3 = 0 => k = -3Everything seems to add up. I think I'm confident that k is indeed -3.**Final Answer**The value of ( k ) is boxed{-3}.

answer：Okay, so Lisa went on a field trip and counted 300 heads and 688 legs among birds, mammals, and reptiles. Birds have 2 legs, mammals have 4 legs, and reptiles have 6 legs. I need to figure out how many birds there were.First, let me define some variables to make this easier. Let's say:- ( b ) = number of birds- ( m ) = number of mammals- ( r ) = number of reptilesSince each animal has one head, the total number of heads gives me the equation:[ b + m + r = 300 ]For the legs, birds have 2 legs, mammals have 4, and reptiles have 6. So the total number of legs is:[ 2b + 4m + 6r = 688 ]Now, I have a system of two equations:1. ( b + m + r = 300 )2. ( 2b + 4m + 6r = 688 )I can try to simplify this system to find the values of ( b ), ( m ), and ( r ). Maybe I can express one variable in terms of the others and substitute.Let me start by simplifying the second equation. If I divide the entire equation by 2, it becomes:[ b + 2m + 3r = 344 ]Now I have:1. ( b + m + r = 300 )2. ( b + 2m + 3r = 344 )If I subtract the first equation from the second, I can eliminate ( b ):[ (b + 2m + 3r) - (b + m + r) = 344 - 300 ][ b + 2m + 3r - b - m - r = 44 ][ m + 2r = 44 ]Okay, so now I have:[ m + 2r = 44 ]This equation relates mammals and reptiles. I can express ( m ) in terms of ( r ):[ m = 44 - 2r ]Now, I can substitute this back into the first equation to find ( b ):[ b + (44 - 2r) + r = 300 ][ b + 44 - r = 300 ][ b = 300 - 44 + r ][ b = 256 + r ]Hmm, so ( b = 256 + r ). But I know that the number of animals can't be negative, so ( r ) must be such that ( m ) is non-negative. From ( m = 44 - 2r ), ( 44 - 2r geq 0 ), which implies ( r leq 22 ).So ( r ) can be at most 22. Let me test ( r = 22 ):- ( m = 44 - 2(22) = 44 - 44 = 0 )- ( b = 256 + 22 = 278 )Wait, but if ( m = 0 ), that means there are no mammals. Let me check if this satisfies the legs equation:[ 2b + 4m + 6r = 2(278) + 4(0) + 6(22) = 556 + 0 + 132 = 688 ]Yes, that works. So with ( r = 22 ), ( m = 0 ), and ( b = 278 ), the numbers add up.But wait, the answer choices don't include 278. The closest is 234. Did I make a mistake?Let me double-check my equations. Maybe I misapplied the substitution.Starting again:From ( m + 2r = 44 ), ( m = 44 - 2r ).Substituting into ( b + m + r = 300 ):[ b + (44 - 2r) + r = 300 ][ b + 44 - r = 300 ][ b = 300 - 44 + r ][ b = 256 + r ]Wait, that's correct. So if ( r = 22 ), ( b = 278 ). But 278 isn't an option. Maybe I need to consider a different value for ( r ).Let me try ( r = 12 ):- ( m = 44 - 2(12) = 44 - 24 = 20 )- ( b = 256 + 12 = 268 )Check legs:[ 2(268) + 4(20) + 6(12) = 536 + 80 + 72 = 688 ]That also works. But 268 isn't an option either.Wait, maybe I need to find a value of ( r ) such that ( b ) is one of the answer choices. The options are 234, 256, 300, 344, 400.Looking at ( b = 256 + r ), if ( b ) is 256, then ( r = 0 ). Let's check:- ( r = 0 )- ( m = 44 - 2(0) = 44 )- ( b = 256 )Check legs:[ 2(256) + 4(44) + 6(0) = 512 + 176 + 0 = 688 ]Yes, that works. So ( b = 256 ) is a possible answer. But 256 is option B, but earlier I got 278 when ( r = 22 ). Why is there a discrepancy?Wait, maybe I misinterpreted the equations. Let me go back.Original equations:1. ( b + m + r = 300 )2. ( 2b + 4m + 6r = 688 )Simplify equation 2 by dividing by 2:[ b + 2m + 3r = 344 ]Subtract equation 1:[ (b + 2m + 3r) - (b + m + r) = 344 - 300 ][ m + 2r = 44 ]So ( m = 44 - 2r ). Then substituting back:[ b + (44 - 2r) + r = 300 ][ b + 44 - r = 300 ][ b = 256 + r ]So ( b = 256 + r ). Therefore, ( b ) must be at least 256, since ( r ) can't be negative.Looking at the answer choices, 256 is an option (B). So that must be the answer.Wait, but earlier when I took ( r = 22 ), I got ( b = 278 ), which also satisfies the equations. So why is 256 the answer?Ah, because the problem doesn't specify that all three types of animals must be present. So if ( r = 0 ), then ( b = 256 ) and ( m = 44 ). That's a valid solution.But the question is asking for the number of birds. Since 256 is an option and it's the minimum number of birds possible, that must be the answer.Wait, but if ( r ) can be up to 22, then ( b ) can be up to 278. But since 256 is an option and it's the smallest possible number of birds, maybe that's the intended answer.Alternatively, maybe I need to consider that the number of reptiles can't be more than a certain number because of the legs.Wait, let me think differently. Maybe I can set up the equations differently.Let me assume that all 300 animals are birds. Then the total number of legs would be ( 2 times 300 = 600 ). But Lisa counted 688 legs, which is 88 more legs than if all were birds.Each mammal has 2 more legs than a bird, and each reptile has 4 more legs than a bird. So the extra legs come from mammals and reptiles.Let me define:- Let ( x ) be the number of mammals.- Let ( y ) be the number of reptiles.Then, the total extra legs are ( 2x + 4y = 88 ).Also, the total number of animals is ( x + y leq 300 ), but since we're replacing birds with mammals and reptiles, ( x + y ) is the number of non-birds.Wait, actually, the total number of animals is still 300, so ( b + x + y = 300 ). But since ( b = 300 - x - y ), we can express everything in terms of ( x ) and ( y ).But maybe this approach is complicating things. Let me stick to the original equations.From earlier, I have:[ b = 256 + r ]And ( r ) can be from 0 to 22. So the number of birds can range from 256 to 278.But since 256 is an option, and it's the minimum number of birds, that must be the answer.Wait, but the problem doesn't specify that all three types of animals are present. So if there are no reptiles, then ( r = 0 ), ( m = 44 ), and ( b = 256 ). That's a valid solution.Alternatively, if there are reptiles, the number of birds increases. But since 256 is an option and it's the smallest possible number, that must be the answer.Wait, but in the first approach, when I took ( r = 22 ), I got ( b = 278 ), which is not an option. So maybe the correct answer is 256.But I'm confused because both solutions satisfy the equations. Maybe I need to consider that the number of mammals can't be negative, so ( m = 44 - 2r geq 0 ), which means ( r leq 22 ). So ( r ) can be 0 to 22.But since the problem doesn't specify that all three types are present, the minimum number of birds is 256, which is an option.Wait, but the answer choices include 234, which is less than 256. How is that possible?Wait, maybe I made a mistake in my earlier reasoning. Let me try another approach.Let me assume that all 300 animals are birds. Then legs = 600. But we have 688 legs, which is 88 more.Each mammal adds 2 extra legs, each reptile adds 4 extra legs.So, let ( x ) be the number of mammals and ( y ) be the number of reptiles.Then:[ 2x + 4y = 88 ][ x + y leq 300 ]But also, the total number of animals is 300, so:[ x + y = 300 - b ]But I'm not sure if this helps. Maybe I can solve for ( x ) and ( y ).From ( 2x + 4y = 88 ), divide by 2:[ x + 2y = 44 ]So ( x = 44 - 2y )Since ( x ) and ( y ) must be non-negative integers, ( y ) can be from 0 to 22.So the number of birds ( b = 300 - x - y = 300 - (44 - 2y) - y = 300 - 44 + y = 256 + y )So ( b = 256 + y ), where ( y ) is from 0 to 22.So the number of birds can be from 256 to 278.But the answer choices are 234, 256, 300, 344, 400.So 256 is an option, which is the minimum number of birds. Therefore, the answer must be 256.Wait, but earlier I thought 234 was possible. Maybe I need to check if 234 is possible.If ( b = 234 ), then ( x + y = 300 - 234 = 66 )From ( x + 2y = 44 ), and ( x + y = 66 )Subtracting the two equations:[ (x + 2y) - (x + y) = 44 - 66 ][ y = -22 ]Which is not possible since ( y ) can't be negative. So 234 is not possible.Therefore, the minimum number of birds is 256, which is an option. So the answer is 256.