answer：Alright, so I have this problem here: The sum of the greatest common divisor (GCD) and the least common multiple (LCM) of two positive integers (a) and (b) is equal to (a + b). I need to prove that one of the numbers (a) or (b) is divisible by the other. Hmm, okay, let's break this down step by step.First, I remember that for any two positive integers, the product of their GCD and LCM is equal to the product of the numbers themselves. So, mathematically, that's:[text{GCD}(a, b) times text{LCM}(a, b) = a times b]Let me denote the GCD of (a) and (b) as (d). So, (d = text{GCD}(a, b)). Then, according to the problem, the sum of the GCD and LCM is equal to (a + b):[d + text{LCM}(a, b) = a + b]From the property I mentioned earlier, I can express the LCM in terms of (a), (b), and (d):[text{LCM}(a, b) = frac{a times b}{d}]Substituting this into the equation from the problem:[d + frac{a times b}{d} = a + b]To make this equation easier to handle, I'll multiply both sides by (d) to eliminate the denominator:[d^2 + a times b = d times (a + b)]Now, let's rearrange the equation to bring all terms to one side:[d^2 - d times (a + b) + a times b = 0]Looking at this quadratic equation in terms of (d), I notice that it might factor nicely. Let me try factoring:[(d - a)(d - b) = 0]So, this equation implies that either (d = a) or (d = b). If (d = a), then (a) divides (b) because the GCD of (a) and (b) is (a), meaning (a) is a divisor of (b). Similarly, if (d = b), then (b) divides (a). Therefore, in either case, one of the numbers (a) or (b) is divisible by the other. That seems to solve the problem!Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the beginning:1. I used the property that (text{GCD}(a, b) times text{LCM}(a, b) = a times b). That's a standard result, so that's correct.2. I set (d = text{GCD}(a, b)) and substituted into the given equation. That seems fine.3. I multiplied both sides by (d) to eliminate the fraction. That's a valid operation since (d) is a positive integer and thus not zero.4. I rearranged the equation and factored it. The factoring step is crucial here. Let me verify that:Starting from:[d^2 - d(a + b) + ab = 0]This is a quadratic in (d). The quadratic can be written as:[d^2 - (a + b)d + ab = 0]Which factors into:[(d - a)(d - b) = 0]Yes, that's correct because:[(d - a)(d - b) = d^2 - (a + b)d + ab]So, my factoring is accurate.5. From the factoring, I concluded that (d = a) or (d = b). That's correct because if a product of two terms is zero, at least one of the terms must be zero.6. Finally, if (d = a), then (a) divides (b), and if (d = b), then (b) divides (a). That makes sense because the GCD being equal to one of the numbers implies that the other number is a multiple of the GCD.So, all my steps seem solid, and I don't see any errors in my reasoning. Therefore, I'm confident that the conclusion is correct: one of the numbers (a) or (b) must be divisible by the other.Just to make sure, let's consider an example. Suppose (a = 6) and (b = 3). Then, (text{GCD}(6, 3) = 3) and (text{LCM}(6, 3) = 6). The sum of GCD and LCM is (3 + 6 = 9), and (a + b = 6 + 3 = 9). So, the condition holds, and indeed, (6) is divisible by (3).Another example: (a = 8) and (b = 4). (text{GCD}(8, 4) = 4), (text{LCM}(8, 4) = 8). Sum is (4 + 8 = 12), and (a + b = 8 + 4 = 12). Again, the condition holds, and (8) is divisible by (4).What if (a) and (b) are equal? Let's say (a = b = 5). Then, (text{GCD}(5, 5) = 5), (text{LCM}(5, 5) = 5). Sum is (5 + 5 = 10), and (a + b = 5 + 5 = 10). Here, both numbers are equal, so each divides the other.What if the numbers are coprime? For example, (a = 2) and (b = 3). Then, (text{GCD}(2, 3) = 1), (text{LCM}(2, 3) = 6). Sum is (1 + 6 = 7), but (a + b = 2 + 3 = 5). So, the condition doesn't hold here, which is consistent with the problem statement because neither 2 divides 3 nor 3 divides 2.This further confirms that the condition in the problem implies that one number must divide the other.I think I've covered all bases here. My reasoning seems solid, and the examples support the conclusion. I'm ready to present the step-by-step explanation.

answer：Okay, so I have this problem about finding the minimum area enclosed by the curve ( y = x^2 ), the lines ( x = 0 ), ( x = 1 ), and ( y = t^2 ), where ( t ) is between 0 and 1. The options are given as A: ( frac{2}{3} ), B: ( frac{1}{3} ), C: ( frac{1}{2} ), and D: ( frac{1}{4} ). I need to figure out which one is the correct answer.First, I should visualize the area we're talking about. The curve ( y = x^2 ) is a parabola opening upwards. The lines ( x = 0 ) and ( x = 1 ) are vertical lines at the y-axis and at ( x = 1 ). The line ( y = t^2 ) is a horizontal line somewhere between ( y = 0 ) and ( y = 1 ) since ( t ) is between 0 and 1.So, the area enclosed by these four boundaries is the region that's under the parabola ( y = x^2 ), above the horizontal line ( y = t^2 ), and between ( x = 0 ) and ( x = 1 ). But wait, actually, depending on the value of ( t ), the horizontal line ( y = t^2 ) will intersect the parabola at some point. Let me find where they intersect.Setting ( y = x^2 ) equal to ( y = t^2 ), we get ( x^2 = t^2 ), so ( x = t ) or ( x = -t ). But since we're only considering ( x ) from 0 to 1, the intersection point is at ( x = t ). So, the area is split into two parts: from ( x = 0 ) to ( x = t ), the top boundary is ( y = t^2 ), and from ( x = t ) to ( x = 1 ), the top boundary is the parabola ( y = x^2 ).Wait, actually, no. Let me think again. If ( y = t^2 ) is a horizontal line, and ( y = x^2 ) is a parabola, then above ( y = t^2 ), the parabola is above the line, and below ( y = t^2 ), the line is above the parabola. So, actually, the area between ( x = 0 ) and ( x = t ) is bounded above by ( y = t^2 ) and below by ( y = x^2 ), and the area between ( x = t ) and ( x = 1 ) is bounded above by ( y = x^2 ) and below by ( y = t^2 ).So, to find the total area, I need to compute two integrals:1. From ( x = 0 ) to ( x = t ), the area between ( y = t^2 ) and ( y = x^2 ).2. From ( x = t ) to ( x = 1 ), the area between ( y = x^2 ) and ( y = t^2 ).So, the total area ( A ) can be expressed as:[A = int_{0}^{t} (t^2 - x^2) , dx + int_{t}^{1} (x^2 - t^2) , dx]Let me compute each integral separately.First integral:[int_{0}^{t} (t^2 - x^2) , dx = left[ t^2 x - frac{x^3}{3} right]_{0}^{t} = left( t^3 - frac{t^3}{3} right) - (0 - 0) = frac{2t^3}{3}]Second integral:[int_{t}^{1} (x^2 - t^2) , dx = left[ frac{x^3}{3} - t^2 x right]_{t}^{1} = left( frac{1}{3} - t^2 right) - left( frac{t^3}{3} - t^3 right) = left( frac{1}{3} - t^2 right) - left( -frac{2t^3}{3} right) = frac{1}{3} - t^2 + frac{2t^3}{3}]So, adding both integrals together:[A = frac{2t^3}{3} + frac{1}{3} - t^2 + frac{2t^3}{3} = frac{4t^3}{3} - t^2 + frac{1}{3}]So, the area as a function of ( t ) is:[A(t) = frac{4}{3}t^3 - t^2 + frac{1}{3}]Now, I need to find the value of ( t ) in the interval ( (0, 1) ) that minimizes this area. To find the minimum, I should take the derivative of ( A(t) ) with respect to ( t ), set it equal to zero, and solve for ( t ).Let's compute the derivative ( A'(t) ):[A'(t) = frac{d}{dt} left( frac{4}{3}t^3 - t^2 + frac{1}{3} right) = 4t^2 - 2t]Set the derivative equal to zero to find critical points:[4t^2 - 2t = 0][2t(2t - 1) = 0]So, the critical points are at ( t = 0 ) and ( t = frac{1}{2} ). Since ( t ) is in the interval ( (0, 1) ), we can ignore ( t = 0 ) and focus on ( t = frac{1}{2} ).Now, to confirm whether this critical point is a minimum, I can use the second derivative test.Compute the second derivative ( A''(t) ):[A''(t) = frac{d}{dt} (4t^2 - 2t) = 8t - 2]Evaluate ( A''(t) ) at ( t = frac{1}{2} ):[A''left( frac{1}{2} right) = 8 times frac{1}{2} - 2 = 4 - 2 = 2]Since ( A''left( frac{1}{2} right) > 0 ), the function has a local minimum at ( t = frac{1}{2} ).Therefore, the minimum area occurs at ( t = frac{1}{2} ). Now, let's compute the area at this value.Substitute ( t = frac{1}{2} ) into ( A(t) ):[Aleft( frac{1}{2} right) = frac{4}{3} left( frac{1}{2} right)^3 - left( frac{1}{2} right)^2 + frac{1}{3}][= frac{4}{3} times frac{1}{8} - frac{1}{4} + frac{1}{3}][= frac{4}{24} - frac{6}{24} + frac{8}{24}][= frac{4 - 6 + 8}{24}][= frac{6}{24}][= frac{1}{4}]So, the minimum area is ( frac{1}{4} ), which corresponds to option D.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, computing ( A(t) ):[A(t) = frac{4}{3}t^3 - t^2 + frac{1}{3}]At ( t = frac{1}{2} ):[frac{4}{3} times left( frac{1}{2} right)^3 = frac{4}{3} times frac{1}{8} = frac{4}{24} = frac{1}{6}][- left( frac{1}{2} right)^2 = -frac{1}{4}][+ frac{1}{3}]So, adding these together:[frac{1}{6} - frac{1}{4} + frac{1}{3}]Convert to common denominator, which is 12:[frac{2}{12} - frac{3}{12} + frac{4}{12} = frac{2 - 3 + 4}{12} = frac{3}{12} = frac{1}{4}]Yes, that's correct. So, the minimum area is indeed ( frac{1}{4} ).Just to make sure, let me consider the endpoints as well. When ( t ) approaches 0, what happens to the area?As ( t to 0^+ ):[A(t) = frac{4}{3}t^3 - t^2 + frac{1}{3} approx 0 - 0 + frac{1}{3} = frac{1}{3}]Similarly, when ( t to 1^- ):[A(t) = frac{4}{3}(1)^3 - (1)^2 + frac{1}{3} = frac{4}{3} - 1 + frac{1}{3} = frac{4}{3} - frac{3}{3} + frac{1}{3} = frac{2}{3}]So, at the endpoints, the area is ( frac{1}{3} ) and ( frac{2}{3} ), which are both larger than ( frac{1}{4} ). Therefore, the minimum must indeed occur at ( t = frac{1}{2} ), giving the area ( frac{1}{4} ).I think that's thorough enough. I don't see any mistakes in my reasoning or calculations.**Final Answer**boxed{D}

answer：Okay, so I have this problem where I need to prove that if I divide the interval [0,1] into p+q equal parts, where p and q are coprime natural numbers, then except for the leftmost and rightmost subintervals, each of the remaining subintervals contains exactly one of the fractions 1/p, 2/p, ..., (p-1)/p or 1/q, 2/q, ..., (q-1)/q. Hmm, that sounds a bit tricky, but let me try to break it down.First, I know that p and q being coprime means their greatest common divisor is 1. That might be important because it could imply that certain fractions won't overlap or something. I also know that dividing [0,1] into p+q equal parts means each subinterval has a length of 1/(p+q). So, the points where the interval is divided are at k/(p+q) for k = 0, 1, 2, ..., p+q.Now, the fractions we're looking at are 1/p, 2/p, ..., (p-1)/p and 1/q, 2/q, ..., (q-1)/q. These are all fractions between 0 and 1, excluding 0 and 1 themselves. So, each of these fractions should lie somewhere within the interval [0,1], and we need to show that each of them falls into one of the middle subintervals, not the first or the last.Let me think about how these fractions relate to the subintervals. Each subinterval is of the form [k/(p+q), (k+1)/(p+q)) for k from 0 to p+q-1. The first subinterval is [0, 1/(p+q)) and the last one is [(p+q-1)/(p+q), 1]. So, I need to show that none of the fractions 1/p, 2/p, ..., (p-1)/p or 1/q, 2/q, ..., (q-1)/q fall into these first or last subintervals, and each falls into exactly one of the remaining subintervals.Maybe I can start by considering the positions of these fractions relative to the division points. Let's take a fraction, say, i/p. I want to see which subinterval it falls into. So, I need to find an integer k such that k/(p+q) ≤ i/p < (k+1)/(p+q). Similarly, for j/q, I need to find k such that k/(p+q) ≤ j/q < (k+1)/(p+q).Wait, maybe I can rearrange these inequalities to find k in terms of i and p. Let's do that for i/p.Starting with k/(p+q) ≤ i/p < (k+1)/(p+q). Multiply all parts by p(p+q) to eliminate denominators:k p ≤ i (p + q) < (k + 1) pSo, k p ≤ i p + i q < (k + 1) pSubtract i p from all parts:k p - i p ≤ i q < (k + 1) p - i pFactor p:p(k - i) ≤ i q < p(k + 1 - i)Hmm, not sure if that helps. Maybe another approach. Since p and q are coprime, perhaps I can use the fact that the fractions i/p and j/q are all distinct. That is, no fraction i/p equals j/q because p and q are coprime. So, each fraction is unique.Also, since p and q are coprime, the number of fractions i/p is p-1 and j/q is q-1, so total p+q-2 fractions, which matches the number of middle subintervals (since there are p+q subintervals in total, minus 2 gives p+q-2). So, that suggests that each fraction must lie in a unique subinterval.But how do I show that each fraction actually lies in one of these middle subintervals? Maybe I can use the concept of Farey sequences or something related to mediants, but I'm not too sure.Alternatively, maybe I can consider the positions of these fractions relative to the division points. Since each subinterval is of length 1/(p+q), and the fractions are spaced at intervals of 1/p and 1/q, which are both larger than 1/(p+q) because p and q are at least 1. Wait, actually, 1/p and 1/q could be larger or smaller depending on p and q. Hmm.Wait, let's think about the spacing. The fractions i/p are spaced 1/p apart, and the subintervals are spaced 1/(p+q) apart. Since p and q are coprime, 1/p and 1/q are incommensurate with 1/(p+q). That might mean that the fractions don't align with the division points, so each fraction must lie strictly within a subinterval.But how do I formalize that? Maybe I can use the fact that since p and q are coprime, the fractions i/p and j/q cannot be equal to any of the division points k/(p+q). Because if i/p = k/(p+q), then (p+q)i = kp, which would imply that p divides (p+q)i. Since p and q are coprime, p divides i. But i is less than p, so that can't happen. Similarly for j/q.Therefore, none of the fractions i/p or j/q can coincide with the division points. So, each fraction must lie strictly inside one of the subintervals. Now, I need to show that they don't lie in the first or last subintervals.For the first subinterval [0, 1/(p+q)), the smallest fraction is 1/p. So, is 1/p less than 1/(p+q)? Let's see: 1/p < 1/(p+q) would imply p+q < p, which is not true because q is at least 1. So, 1/p is actually greater than 1/(p+q). Therefore, 1/p does not lie in the first subinterval.Similarly, the largest fraction is (p-1)/p. Is (p-1)/p greater than (p+q-1)/(p+q)? Let's check: (p-1)/p > (p+q-1)/(p+q). Cross-multiplying: (p-1)(p+q) > p(p+q-1). Expanding both sides: p(p+q) - (p+q) > p(p+q) - p. Simplify: - (p+q) > -p, which is -p - q > -p, which simplifies to -q > 0, which is false because q is positive. Therefore, (p-1)/p < (p+q-1)/(p+q). So, (p-1)/p does not lie in the last subinterval.Similarly, for the fractions j/q, the smallest is 1/q, which is greater than 1/(p+q) because p+q > q (since p is at least 1). So, 1/q does not lie in the first subinterval. The largest fraction is (q-1)/q, which is less than (p+q-1)/(p+q) because (q-1)/q = 1 - 1/q and (p+q-1)/(p+q) = 1 - 1/(p+q). Since 1/q > 1/(p+q), 1 - 1/q < 1 - 1/(p+q). Therefore, (q-1)/q < (p+q-1)/(p+q), so it doesn't lie in the last subinterval.So, all the fractions i/p and j/q lie strictly inside the middle subintervals. Now, I need to show that each subinterval contains exactly one fraction. Since there are p+q-2 fractions and p+q-2 middle subintervals, it's sufficient to show that each fraction lies in a unique subinterval.Given that p and q are coprime, the fractions i/p and j/q are all distinct. So, no two fractions are the same. Also, since the subintervals are of length 1/(p+q), and the spacing between consecutive fractions is either 1/p or 1/q, which are both larger than 1/(p+q), it's possible that each fraction lies in a separate subinterval.Wait, actually, the spacing between consecutive fractions is not necessarily larger than 1/(p+q). For example, if p=2 and q=3, then 1/2 and 1/3 are spaced by 1/6, which is equal to 1/(p+q) in this case. Hmm, so maybe that approach isn't correct.Alternatively, perhaps I can use the fact that the fractions i/p and j/q are all distinct and lie in the open interval (0,1), and the subintervals partition [0,1] into p+q intervals, so each fraction must lie in exactly one subinterval. Since there are p+q-2 fractions and p+q-2 middle subintervals, each subinterval must contain exactly one fraction.But I need to make sure that no two fractions lie in the same subinterval. Suppose, for contradiction, that two fractions, say i/p and j/q, lie in the same subinterval [k/(p+q), (k+1)/(p+q)). Then, |i/p - j/q| < 1/(p+q). Let's see if that's possible.Compute |i/p - j/q| = |(iq - jp)/(pq)|. Since p and q are coprime, the numerator iq - jp must be at least 1 in absolute value because otherwise, if iq = jp, then p divides iq, but p and q are coprime, so p divides i, which is less than p, so i=0, which is not in our set. Similarly, j=0, which is not in our set. Therefore, |iq - jp| ≥ 1. Therefore, |i/p - j/q| ≥ 1/(pq). Now, since p and q are at least 1, 1/(pq) is at least 1/(p+q) only if p and q are both 1, but since they are coprime and natural numbers, p and q can't both be 1 unless p=q=1, but then p+q=2, and the subintervals would be of length 1/2, but the fractions would be none since p-1=0 and q-1=0. So, in general, 1/(pq) > 1/(p+q) because pq > p+q for p,q ≥2. Therefore, |i/p - j/q| ≥ 1/(pq) > 1/(p+q), which implies that two fractions cannot lie in the same subinterval. Therefore, each fraction must lie in a unique subinterval.Thus, we've shown that each of the p+q-2 fractions lies in exactly one of the p+q-2 middle subintervals, and no two fractions lie in the same subinterval. Therefore, each middle subinterval contains exactly one of the fractions 1/p, 2/p, ..., (p-1)/p, 1/q, 2/q, ..., (q-1)/q.I think that covers it. I started by considering the positions of the fractions relative to the subintervals, used the fact that p and q are coprime to show that the fractions are distinct and don't coincide with the division points, and then showed that each fraction must lie in a unique subinterval by considering the minimum distance between any two fractions. This ensures that each middle subinterval contains exactly one fraction.

answer：First, I need to determine the total number of ways to arrange the six numbers without any restrictions. Since there are two identical "1"s, the total number of unique arrangements is calculated by dividing the factorial of the total number of items by the factorial of the number of identical items:[frac{6!}{2!} = frac{720}{2} = 360]Next, I consider the restriction that the two "1"s must be adjacent. To simplify the problem, I treat the two "1"s as a single entity. This reduces the number of items to arrange from six to five: the "11" bundle, 2, 3, 5, and 8. The number of ways to arrange these five items is:[5! = 120]Therefore, the number of different numerical passwords that can be set, considering the condition that the two "1"s must be adjacent, is:[boxed{120}]Hence, the answer is: boxed{text{A}}.

answer：Okay, so I have this geometry problem here about three lines, a, b, and c. They're pairwise parallel, which means each pair of lines is parallel to each other. But they're also not coplanar, meaning all three lines don't lie on the same plane. The question is asking how many planes these three lines can determine, which is m, and how many parts these planes divide the space into, which is n. The options are A: m=2, n=2; B: m=2, n=6; C: m=3, n=7; D: m=3, n=8.Alright, let's start by understanding what it means for three lines to be pairwise parallel and not coplanar. If they are pairwise parallel, each line is parallel to the other two. But since they are not coplanar, they can't all lie on the same flat surface. So, they must be arranged in space in such a way that each pair is parallel, but the three together aren't on a single plane.Now, the first part is figuring out how many planes these three lines determine. Each pair of parallel lines should lie on a unique plane, right? Because two parallel lines define a plane. So, for three lines, how many pairs do we have? Let's see, line a and b, line a and c, and line b and c. That's three pairs. So, each pair defines a plane, so that would be three planes. So, m should be 3. That rules out options A and B because they say m=2.Now, onto the second part: how many parts do these three planes divide the space into? Hmm, this is a bit trickier. I remember that when you have multiple planes in space, the number of regions they divide the space into can be calculated with a formula. I think the formula is similar to how lines divide a plane, but extended into three dimensions.Let me recall. For n planes in three-dimensional space, the maximum number of regions they can divide the space into is given by the formula:R(n) = (n^3 + 5n + 6)/6Wait, is that right? Or is it R(n) = (n^3 + 5n + 6)/6? Hmm, no, that doesn't sound quite right. Let me think again. I think it's R(n) = (n^3 + 5n + 6)/6, but maybe I'm mixing it up.Wait, no, actually, for regions divided by planes, the formula is similar to the one for lines dividing a plane but extended. For lines in a plane, the formula is R(n) = (n^2 + n + 2)/2. For planes in space, it's R(n) = (n^3 + 5n + 6)/6. Let me check with n=1: one plane divides space into two regions. Plugging into the formula: (1 + 5 + 6)/6 = 12/6=2. That's correct.For n=2: two planes intersecting along a line divide space into four regions. Plugging into the formula: (8 + 10 + 6)/6=24/6=4. Correct again.For n=3: three planes. If they're all intersecting each other, the maximum number of regions is 8. But wait, in our case, the planes are determined by three parallel lines. So, are these planes intersecting each other?Wait, each plane is determined by a pair of parallel lines. Since the lines are all parallel, each plane is parallel to the others? No, wait, no. If the lines are pairwise parallel but not coplanar, then each pair of lines defines a plane, but these planes aren't necessarily parallel.Wait, hold on. If three lines are pairwise parallel but not coplanar, then each pair of lines defines a plane, but these planes must intersect each other because the lines are not all in the same plane.Wait, let me visualize this. Imagine three parallel lines, but they're not all on the same plane. So, line a is on plane 1, line b is on plane 2, and line c is on plane 3. But since lines a and b are parallel, they lie on plane 1. Similarly, lines a and c lie on plane 2, and lines b and c lie on plane 3. So, each plane contains two of the lines.But since the lines are not coplanar, these planes must intersect each other. Each pair of planes will intersect along a line. Specifically, plane 1 (containing a and b) and plane 2 (containing a and c) intersect along line a. Similarly, plane 1 and plane 3 intersect along line b, and plane 2 and plane 3 intersect along line c.So, we have three planes, each intersecting the other two along distinct lines. So, these three planes are like three pages of a book, each intersecting along a different edge. So, how many regions do these three planes divide the space into?I think the formula for regions divided by planes is R(n) = R(n-1) + n. Wait, no, that's for lines in a plane. For planes in space, the formula is R(n) = R(n-1) + n(n - 1)/2 + 1? Hmm, maybe not. Let me think again.Wait, actually, the formula for the maximum number of regions divided by n planes is R(n) = (n^3 + 5n + 6)/6. So, for n=3, that would be (27 + 15 + 6)/6 = 48/6=8. So, 8 regions. But wait, in our case, are these three planes in general position? Or are they intersecting along lines that are themselves parallel?Wait, no, in our case, each pair of planes intersects along a line, but those lines are the original lines a, b, and c, which are parallel. So, the lines of intersection of the planes are parallel. So, the planes are not in general position because their lines of intersection are parallel, meaning they don't intersect each other in a single point.So, does that affect the number of regions? Because if the planes are in general position, meaning each pair intersects along a distinct line, and no three planes intersect along a common line, then the formula R(n) = (n^3 + 5n + 6)/6 gives the maximum number of regions.But in our case, the three planes intersect along three parallel lines, so they are not in general position. Therefore, the number of regions might be less than 8.Wait, let me think about it. If I have three planes, each pair intersecting along a parallel line, how does that divide the space?Imagine three planes, each pair intersecting along a line, and all those lines are parallel. So, it's like three pages of a book, but instead of fanning out from a single spine, each pair of pages intersects along a different spine, but all those spines are parallel.So, how many regions does that create?Let me try to visualize it step by step.First, with one plane, we have two regions.Adding a second plane, which intersects the first along a line. So, now we have four regions.Adding a third plane, which intersects each of the first two planes along a different line, but both lines are parallel to each other.So, how does this third plane divide the space?Each time a new plane is added, it can potentially divide existing regions into more regions. The number of new regions created by the nth plane is equal to the number of regions it intersects, which is equal to the number of intersections it has with the previous planes.In this case, the third plane intersects the first two planes along two parallel lines. So, the third plane is intersected by two lines, which are parallel, so it's like the third plane has two parallel lines on it.How does that divide the third plane? Each line divides the plane into two parts, but since they're parallel, they divide the plane into three parts.But how does that translate into regions in space?Wait, maybe it's better to think about how many times the third plane is intersected by the existing arrangement.Each existing plane can intersect the new plane along a line. Since the lines of intersection are parallel, the third plane is intersected by two parallel lines.So, in the third plane, these two parallel lines divide it into three regions. Each of these regions corresponds to a new region in space.So, the third plane would add three new regions.So, starting with four regions after two planes, adding the third plane would add three regions, making a total of seven regions.Wait, but I thought the formula gave eight regions for three planes in general position.So, in our case, since the planes are not in general position, the number of regions is less. It's seven instead of eight.So, n=7.Therefore, m=3 and n=7, which is option C.Wait, but let me double-check. Maybe my reasoning is flawed.Alternatively, maybe I can think of it as each plane being added and how many regions it creates.First plane: splits space into two.Second plane: intersects the first along a line, so it adds two more regions, total four.Third plane: intersects each of the first two planes along a line, but these lines are parallel. So, how many regions does the third plane add?In general, when adding the nth plane, the maximum number of new regions it can create is n. But in our case, since the intersections are parallel, maybe it's less.Wait, no, actually, the number of regions a new plane can add is equal to the number of times it intersects the previous arrangement, which is the number of lines it intersects.Each intersection with a previous plane adds a line on the new plane, which can divide existing regions.But in our case, the third plane intersects the first two planes along two parallel lines. So, on the third plane, these two lines divide it into three regions. Each of these regions corresponds to cutting through existing space regions, thereby creating new regions.So, the third plane would add three regions, going from four to seven.Yes, that seems right.Alternatively, maybe I can think of it as the arrangement of three planes, each pair intersecting along parallel lines, forming a sort of prism-like structure, dividing space into seven regions.So, yeah, I think n=7.Therefore, the answer should be C: m=3, n=7.**Final Answer**boxed{C}

answer：Okay, so I have this problem here: I need to find the length of the chord intercepted by the line ( x - y - 5 = 0 ) on the circle given by the equation ( x^2 + y^2 - 4x + 4y + 6 = 0 ). Hmm, okay, let's break this down step by step.First, I remember that to find the length of a chord in a circle when you have a line intersecting it, you can use the formula involving the radius of the circle and the distance from the center of the circle to the line. The formula is something like ( 2sqrt{r^2 - d^2} ), where ( r ) is the radius and ( d ) is the distance from the center to the line. So, I need to find both the radius of the circle and the distance from the center to the given line.But before that, I think I need to rewrite the equation of the circle in standard form. The given equation is ( x^2 + y^2 - 4x + 4y + 6 = 0 ). To convert this into the standard form ( (x - h)^2 + (y - k)^2 = r^2 ), I need to complete the square for both the ( x ) and ( y ) terms.Starting with the ( x ) terms: ( x^2 - 4x ). To complete the square, I take half of the coefficient of ( x ), which is ( -4 ), so half of that is ( -2 ), and then square it, getting ( 4 ). So, I can write ( x^2 - 4x ) as ( (x - 2)^2 - 4 ).Similarly, for the ( y ) terms: ( y^2 + 4y ). Half of 4 is 2, and squaring that gives 4. So, ( y^2 + 4y ) can be written as ( (y + 2)^2 - 4 ).Now, substituting these back into the original equation:( (x - 2)^2 - 4 + (y + 2)^2 - 4 + 6 = 0 )Simplifying this:( (x - 2)^2 + (y + 2)^2 - 4 - 4 + 6 = 0 )( (x - 2)^2 + (y + 2)^2 - 2 = 0 )Adding 2 to both sides:( (x - 2)^2 + (y + 2)^2 = 2 )Okay, so now the equation is in standard form. From this, I can see that the center of the circle is at ( (2, -2) ) and the radius ( r ) is ( sqrt{2} ). Got that.Next, I need to find the distance ( d ) from the center of the circle ( (2, -2) ) to the line ( x - y - 5 = 0 ). The formula for the distance from a point ( (x_0, y_0) ) to the line ( ax + by + c = 0 ) is ( d = frac{|ax_0 + by_0 + c|}{sqrt{a^2 + b^2}} ).So, let's identify ( a ), ( b ), and ( c ) from the line equation. The line is ( x - y - 5 = 0 ), so ( a = 1 ), ( b = -1 ), and ( c = -5 ).Plugging the center coordinates into the distance formula:( d = frac{|1*2 + (-1)*(-2) + (-5)|}{sqrt{1^2 + (-1)^2}} )Calculating the numerator:( |2 + 2 - 5| = | -1 | = 1 )Denominator:( sqrt{1 + 1} = sqrt{2} )So, ( d = frac{1}{sqrt{2}} ). Hmm, that can be rationalized as ( frac{sqrt{2}}{2} ). I think that's a nicer way to write it.Now, going back to the chord length formula: ( 2sqrt{r^2 - d^2} ). Let's compute ( r^2 ) and ( d^2 ).( r^2 = (sqrt{2})^2 = 2 )( d^2 = left( frac{sqrt{2}}{2} right)^2 = frac{2}{4} = frac{1}{2} )So, ( r^2 - d^2 = 2 - frac{1}{2} = frac{3}{2} )Taking the square root of that:( sqrt{frac{3}{2}} = frac{sqrt{6}}{2} )Then, multiplying by 2 to get the chord length:( 2 * frac{sqrt{6}}{2} = sqrt{6} )So, the length of the chord is ( sqrt{6} ). Let me just double-check my steps to make sure I didn't make any mistakes.1. Converted the circle equation to standard form correctly: Yes, completed the square properly and simplified to ( (x - 2)^2 + (y + 2)^2 = 2 ).2. Identified center and radius correctly: Center at (2, -2), radius ( sqrt{2} ). That seems right.3. Applied the distance formula correctly: Plugged in (2, -2) into the line equation, calculated the absolute value correctly, and simplified to ( frac{sqrt{2}}{2} ). That looks good.4. Applied the chord length formula: Calculated ( r^2 - d^2 ) correctly as ( 2 - 0.5 = 1.5 ), took square root to get ( sqrt{1.5} = sqrt{6}/2 ), then multiplied by 2 to get ( sqrt{6} ). That seems solid.I think I'm confident with this answer. The chord length is ( sqrt{6} ).**Final Answer**The length of the chord is boxed{sqrt{6}}.