💡Okay, so I have this problem where I need to find the probability that the sum of four randomly selected prime numbers from the first fifteen is even. Hmm, let me break this down step by step.First, I should recall what the first fifteen prime numbers are. Starting from the smallest prime, which is 2, then 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Let me count them to make sure: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Yep, that's fifteen primes.Now, the problem says we're selecting four of these primes without replacement, and we want the sum of these four to be even. I need to figure out the probability of that happening.I remember that the sum of numbers is even if there are an even number of odd numbers in the sum. But wait, primes are mostly odd except for 2. So, let me think about this. If I include 2 in my selection, which is the only even prime, then the sum will depend on how many odd primes I pick along with it.Let me recall: even + odd = odd, and odd + odd = even. So, if I have one even number (which is 2) and three odd numbers, the sum will be even + odd + odd + odd. Let's compute that: even + odd is odd, then odd + odd is even, and even + odd is odd. So, the total sum would be odd. Hmm, that's not what we want.Wait, maybe I should think differently. The sum of four numbers is even if there are an even number of odd numbers in the sum. Since all primes except 2 are odd, the number of odd primes in the selection will determine the parity of the sum.So, if I pick four primes, the sum will be even if there are 0, 2, or 4 odd primes in the selection. But wait, 0 odd primes would mean all four are even, but there's only one even prime, which is 2. So, it's impossible to have four even primes because we only have one. So, 0 is out of the question.Similarly, 4 odd primes would mean all four are odd, which is possible because there are fourteen odd primes in the first fifteen. So, the sum would be the sum of four odd numbers. Now, the sum of four odd numbers: each odd number is 2k + 1, so adding four of them would be 2(k1 + k2 + k3 + k4) + 4, which is 2*(something) + 4, which is even. So, four odd primes would give an even sum.Similarly, if I have two odd primes and two even primes, but wait, we only have one even prime. So, we can't have two even primes. Therefore, the only way to get an even sum is to have either four odd primes or zero even primes, but since zero even primes isn't possible, it's only four odd primes.Wait, hold on. Let me verify that. If I have four odd primes, their sum is even. If I have three odd primes and one even prime (which is 2), then the sum would be odd because 2 is even, and three odds add up to odd, so even + odd is odd. So, that's not good.If I have two odd primes and two even primes, but as I said, we only have one even prime, so that's not possible. Similarly, one odd prime and three even primes is impossible because we only have one even prime. So, the only way to get an even sum is to have four odd primes.Therefore, the number of favorable outcomes is the number of ways to choose four odd primes from the available odd primes. Since there are fifteen primes in total and one of them is even (2), there are fourteen odd primes.So, the number of ways to choose four odd primes is C(14,4). The total number of ways to choose any four primes from the fifteen is C(15,4). Therefore, the probability is C(14,4)/C(15,4).Let me compute these combinations. C(n,k) is n! / (k!(n - k)!).First, C(14,4) is 14! / (4! * 10!) = (14 × 13 × 12 × 11) / (4 × 3 × 2 × 1) = (14 × 13 × 12 × 11) / 24.Let me compute that: 14 × 13 = 182, 182 × 12 = 2184, 2184 × 11 = 24024. Then, 24024 / 24 = 1001.Next, C(15,4) is 15! / (4! * 11!) = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1) = (15 × 14 × 13 × 12) / 24.Compute that: 15 × 14 = 210, 210 × 13 = 2730, 2730 × 12 = 32760. Then, 32760 / 24 = 1365.So, the probability is 1001 / 1365. Now, let's simplify this fraction. Let's see if 1001 and 1365 have a common divisor.I know that 1001 is 7 × 11 × 13. Let me check 1365: 1365 divided by 5 is 273, which is 3 × 91, and 91 is 7 × 13. So, 1365 is 5 × 3 × 7 × 13.So, both numerator and denominator have 7 × 13 as common factors. Let's compute the GCD. The GCD of 1001 and 1365 is 7 × 13 = 91.So, divide numerator and denominator by 91: 1001 ÷ 91 = 11, and 1365 ÷ 91 = 15. So, the simplified fraction is 11/15.Therefore, the probability that the sum of the four selected numbers is even is 11/15.

💡Okay, so I have this problem where I need to find the maximum and minimum values of the sum of squares of ten positive integers, given that their total sum is 2005. Hmm, let me think about how to approach this.First, I remember that for a given sum, the sum of squares is minimized when the numbers are as equal as possible. On the other hand, the sum of squares is maximized when the numbers are as unequal as possible, meaning one number is as large as it can be while the others are as small as possible.Let me start with the minimum value. If I want the numbers to be as equal as possible, I should divide 2005 by 10. Let me calculate that: 2005 divided by 10 is 200.5. Since we're dealing with integers, some numbers will be 200 and some will be 201. Specifically, since 2005 is 5 more than 200*10, five of the numbers will be 201 and the other five will be 200. So, the minimum sum of squares would be 5*(200^2) + 5*(201^2). Let me compute that:200 squared is 40,000, and 201 squared is 40,401. So, 5*40,000 is 200,000, and 5*40,401 is 202,005. Adding those together gives 200,000 + 202,005 = 402,005. Okay, that seems right for the minimum. Now, for the maximum sum of squares, I need to make the numbers as unequal as possible. Since all numbers are positive integers, the smallest possible value for each is 1. If I set nine of the numbers to 1, then the tenth number will be 2005 - 9*1 = 1996. So, the maximum sum of squares would be 9*(1^2) + (1996)^2. Calculating that, 1 squared is 1, so 9*1 is 9. 1996 squared is... let me compute that. 2000 squared is 4,000,000, so 1996 is 4 less. Using the formula (a - b)^2 = a^2 - 2ab + b^2, so (2000 - 4)^2 = 2000^2 - 2*2000*4 + 4^2 = 4,000,000 - 16,000 + 16 = 3,984,016. Adding the 9 from the other terms, the total sum is 3,984,016 + 9 = 3,984,025. Wait, let me double-check that. If I have nine 1s and one 1996, the sum is indeed 9 + 1996 = 2005, which matches the given total. And the squares add up correctly as well. So, I think that's correct.Just to make sure I didn't make any calculation errors, especially with 1996 squared. Let me compute 1996*1996 step by step. 1996 * 1996:First, compute 2000 * 2000 = 4,000,000.Then, subtract 4*2000 + 4*2000 - 4*4. Wait, that might not be the right way. Alternatively, break it down:1996 * 1996 = (2000 - 4)^2 = 2000^2 - 2*2000*4 + 4^2 = 4,000,000 - 16,000 + 16 = 3,984,016. Yeah, that's correct.So, adding 9 gives 3,984,025. Therefore, the minimum sum of squares is 402,005 and the maximum is 3,984,025.**Final Answer**The maximum value is boxed{3984025} and the minimum value is boxed{402005}.

💡Okay, so I have this math expression to evaluate, and I need to figure it out step by step. The expression is:(3/2) × [2 2/3 × (1.875 - 5/6)] ÷ [(0.875 + 1 5/6) ÷ 3 1/4]Hmm, that looks a bit complicated with all the fractions and mixed numbers. I think I should start by breaking it down into smaller parts. Maybe I can handle the operations inside the brackets first.First, let me look at the first part inside the brackets: 2 2/3 × (1.875 - 5/6). I see there's a mixed number and some decimals and fractions. I think it might be easier if I convert all these to fractions so I can work with them more easily.Starting with 2 2/3. That's a mixed number, right? So to convert that to an improper fraction, I multiply the whole number by the denominator and add the numerator. So, 2 × 3 = 6, plus 2 is 8, so 2 2/3 is 8/3.Next, 1.875 is a decimal. I remember that 0.875 is 7/8, so 1.875 should be 1 and 7/8, which is 15/8 as an improper fraction. Let me check: 1 × 8 = 8, plus 7 is 15, so yes, 15/8.Then, 5/6 is already a fraction, so I don't need to convert that. So inside the parentheses, I have 15/8 - 5/6. To subtract these, I need a common denominator. The denominators are 8 and 6. The least common multiple of 8 and 6 is 24. So I'll convert both fractions to have 24 as the denominator.15/8 becomes (15 × 3)/(8 × 3) = 45/24.5/6 becomes (5 × 4)/(6 × 4) = 20/24.Now, subtracting them: 45/24 - 20/24 = 25/24.Okay, so the expression inside the first bracket simplifies to 25/24. Now, I need to multiply that by 8/3 (which was 2 2/3). So, 8/3 × 25/24.Multiplying the numerators: 8 × 25 = 200.Multiplying the denominators: 3 × 24 = 72.So, 200/72. I can simplify that. Both numerator and denominator are divisible by 8. 200 ÷ 8 = 25, and 72 ÷ 8 = 9. So, 25/9.Alright, so the first part inside the brackets simplifies to 25/9.Now, moving on to the denominator part: [(0.875 + 1 5/6) ÷ 3 1/4]. Again, there are decimals and mixed numbers here, so I'll convert them to fractions.0.875 is 7/8, as I remembered earlier.1 5/6 is a mixed number. Converting that to an improper fraction: 1 × 6 = 6, plus 5 is 11, so 11/6.3 1/4 is another mixed number. Converting that: 3 × 4 = 12, plus 1 is 13, so 13/4.So, inside the denominator's brackets, I have 7/8 + 11/6. Again, I need a common denominator. 8 and 6 have an LCM of 24.7/8 becomes (7 × 3)/(8 × 3) = 21/24.11/6 becomes (11 × 4)/(6 × 4) = 44/24.Adding them together: 21/24 + 44/24 = 65/24.Now, I need to divide that by 13/4. So, 65/24 ÷ 13/4. Dividing by a fraction is the same as multiplying by its reciprocal, so 65/24 × 4/13.Multiplying numerators: 65 × 4 = 260.Multiplying denominators: 24 × 13 = 312.So, 260/312. Let me simplify that. Both are divisible by 4: 260 ÷ 4 = 65, 312 ÷ 4 = 78. So, 65/78. Hmm, can this be simplified further? Let's see, 65 and 78 are both divisible by 13. 65 ÷ 13 = 5, 78 ÷ 13 = 6. So, 5/6.Okay, so the denominator part simplifies to 5/6.Now, putting it all back into the original expression: (3/2) × (25/9) ÷ (5/6).Dividing by 5/6 is the same as multiplying by 6/5, so the expression becomes (3/2) × (25/9) × (6/5).Let me multiply these fractions together. Multiply the numerators: 3 × 25 × 6 = 450.Multiply the denominators: 2 × 9 × 5 = 90.So, 450/90. Simplifying that, both are divisible by 90: 450 ÷ 90 = 5, and 90 ÷ 90 = 1. So, the result is 5.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the beginning, converting all to fractions, handling the operations step by step, simplifying where possible. It seems correct. Each step checks out, so I think the final answer is indeed 5.

💡First, I need to determine the man's walking speed in kilometers per hour. I know the length of the bridge is 2500 meters and it takes him 15 minutes to cross it.To calculate the speed, I should convert the time from minutes to hours. There are 60 minutes in an hour, so 15 minutes is 15 divided by 60, which equals 0.25 hours.Next, I'll convert the distance from meters to kilometers. Since 1 kilometer is 1000 meters, 2500 meters is 2500 divided by 1000, which equals 2.5 kilometers.Now, I can use the formula for speed, which is distance divided by time. Plugging in the numbers: 2.5 kilometers divided by 0.25 hours equals 10 kilometers per hour.Therefore, the man's walking speed is 10 kilometers per hour.

💡Okay, so I have this problem where I need to find the equation of a circle that is tangent to another circle C at the point (1,2) and has a radius of (5√2)/2. The given circle C has the equation x² + y² - 6x - 8y + 17 = 0. Hmm, let me try to figure this out step by step.First, I think I should rewrite the equation of circle C in standard form so I can easily identify its center and radius. To do that, I need to complete the squares for both x and y terms.Starting with the x terms: x² - 6x. To complete the square, I take half of -6, which is -3, and square it to get 9. So, x² - 6x becomes (x - 3)² - 9.Similarly, for the y terms: y² - 8y. Half of -8 is -4, and squaring that gives 16. So, y² - 8y becomes (y - 4)² - 16.Now, substituting these back into the equation:(x - 3)² - 9 + (y - 4)² - 16 + 17 = 0Simplify the constants: -9 -16 +17 = -8So, the equation becomes:(x - 3)² + (y - 4)² = 8Okay, so circle C has center at (3,4) and radius √8, which simplifies to 2√2.Now, I need to find another circle that is tangent to this circle at (1,2) and has a radius of (5√2)/2. Let me denote the center of the desired circle as (h,k). Since the circle passes through (1,2), the distance from (h,k) to (1,2) must be equal to the radius, which is (5√2)/2.So, the distance formula gives:√[(h - 1)² + (k - 2)²] = (5√2)/2Squaring both sides:(h - 1)² + (k - 2)² = (25 * 2)/4 = 50/4 = 12.5So, equation (1): (h - 1)² + (k - 2)² = 12.5Also, since the two circles are tangent at (1,2), the point (1,2) lies on both circles, and the centers of the two circles and the point of tangency are colinear. That means the line connecting the centers (3,4) and (h,k) must pass through (1,2).So, the slope of the line connecting (3,4) and (h,k) should be the same as the slope of the line connecting (3,4) and (1,2).Calculating the slope between (3,4) and (1,2):Slope = (2 - 4)/(1 - 3) = (-2)/(-2) = 1So, the slope is 1. Therefore, the line connecting (3,4) and (h,k) has a slope of 1. This gives us another equation.The slope between (3,4) and (h,k) is (k - 4)/(h - 3) = 1So, equation (2): k - 4 = h - 3 => k = h + 1Now, I have two equations:1. (h - 1)² + (k - 2)² = 12.52. k = h + 1Let me substitute equation (2) into equation (1):(h - 1)² + ( (h + 1) - 2 )² = 12.5Simplify the second term:(h + 1 - 2) = (h - 1)So, equation becomes:(h - 1)² + (h - 1)² = 12.5Which is:2*(h - 1)² = 12.5Divide both sides by 2:(h - 1)² = 6.25Take square roots:h - 1 = ±√6.25 = ±2.5So, h = 1 ± 2.5Therefore, h = 3.5 or h = -1.5Now, using equation (2): k = h + 1If h = 3.5, then k = 3.5 + 1 = 4.5If h = -1.5, then k = -1.5 + 1 = -0.5So, the centers of the two possible circles are (3.5, 4.5) and (-1.5, -0.5)Now, let me write the equations of these circles.For center (3.5, 4.5):(x - 3.5)² + (y - 4.5)² = (5√2/2)²Calculating the radius squared:(5√2/2)² = (25*2)/4 = 50/4 = 12.5So, equation is:(x - 3.5)² + (y - 4.5)² = 12.5Similarly, for center (-1.5, -0.5):(x + 1.5)² + (y + 0.5)² = 12.5But wait, I need to check if both these circles are tangent to circle C at (1,2). Let me verify.First, for the center (3.5, 4.5):The distance between (3,4) and (3.5,4.5) is √[(3.5 - 3)² + (4.5 - 4)²] = √[(0.5)² + (0.5)²] = √(0.25 + 0.25) = √0.5 ≈ 0.707But the sum of the radii of circle C and the desired circle is 2√2 + (5√2)/2 = (4√2 + 5√2)/2 = (9√2)/2 ≈ 6.364But the distance between centers is only ≈0.707, which is much less than the sum of radii, which suggests they intersect at two points, not tangent. Hmm, that's a problem.Wait, maybe I made a mistake. Let me think again.Actually, when two circles are tangent, the distance between their centers is equal to the sum or difference of their radii. Since the desired circle is tangent to circle C at (1,2), which is a single point, so the distance between centers should be equal to the sum or difference of radii.Given that the desired circle has a larger radius (5√2)/2 ≈ 3.535, and circle C has radius 2√2 ≈ 2.828, so the desired circle is larger. Therefore, the distance between centers should be equal to the difference of radii if they are internally tangent, or sum if externally tangent.But in this case, since the point (1,2) is on both circles, and the desired circle is larger, I think it's externally tangent. So, the distance between centers should be equal to the sum of radii.Wait, but let me calculate the distance between (3,4) and (3.5,4.5):√[(3.5 - 3)^2 + (4.5 - 4)^2] = √[(0.5)^2 + (0.5)^2] = √(0.25 + 0.25) = √0.5 ≈ 0.707Sum of radii: 2√2 + (5√2)/2 = (4√2 + 5√2)/2 = (9√2)/2 ≈ 6.364But 0.707 ≈ √0.5 is much less than 6.364, so they can't be externally tangent. So, maybe they are internally tangent? Then the distance between centers should be equal to the difference of radii.Difference of radii: (5√2)/2 - 2√2 = (5√2 - 4√2)/2 = √2/2 ≈ 0.707Ah, that matches the distance between centers! So, the distance between centers is √0.5 ≈ 0.707, which is equal to √2/2 ≈ 0.707. So, yes, they are internally tangent.Therefore, the circle with center (3.5,4.5) is internally tangent to circle C at (1,2).Similarly, let's check the other center (-1.5, -0.5):Distance between (3,4) and (-1.5, -0.5):√[(-1.5 - 3)^2 + (-0.5 - 4)^2] = √[(-4.5)^2 + (-4.5)^2] = √[20.25 + 20.25] = √40.5 ≈ 6.364Which is equal to the sum of radii: 2√2 + (5√2)/2 = (4√2 + 5√2)/2 = 9√2/2 ≈ 6.364So, this circle is externally tangent to circle C.But the problem states that the circle is tangent at (1,2). So, both circles pass through (1,2). Let me check if (1,2) lies on both circles.For the first circle with center (3.5,4.5):Distance from (3.5,4.5) to (1,2):√[(1 - 3.5)^2 + (2 - 4.5)^2] = √[(-2.5)^2 + (-2.5)^2] = √[6.25 + 6.25] = √12.5 ≈ 3.535, which is equal to the radius (5√2)/2 ≈ 3.535. So, yes, (1,2) is on this circle.Similarly, for the second circle with center (-1.5, -0.5):Distance from (-1.5, -0.5) to (1,2):√[(1 + 1.5)^2 + (2 + 0.5)^2] = √[(2.5)^2 + (2.5)^2] = √[6.25 + 6.25] = √12.5 ≈ 3.535, which is also equal to the radius. So, (1,2) is on both circles.But the problem says "the circle that is tangent to the circle C at (1,2)". So, both circles satisfy this condition. However, the problem might expect both solutions, but perhaps only one is valid based on the position.But let me think again. The circle with center (3.5,4.5) is internally tangent, meaning it's inside circle C, but since it's larger, that might not make sense. Wait, circle C has radius 2√2 ≈ 2.828, and the desired circle has radius ≈3.535, which is larger. So, if it's internally tangent, the desired circle would enclose circle C, but since it's only tangent at one point, it's possible.Alternatively, the externally tangent circle is outside circle C, but still tangent at (1,2). So, both are possible.But the problem says "the circle", implying a single answer. Maybe I need to check which one is correct.Wait, perhaps I made a mistake in assuming both are possible. Let me think about the geometry.Circle C has center (3,4) and radius ≈2.828. The point (1,2) is on circle C. The desired circle has a larger radius, so it must enclose circle C if it's internally tangent, but since it's only tangent at (1,2), it's possible.Alternatively, the externally tangent circle would lie outside circle C, but still tangent at (1,2). So, both are possible.But the problem doesn't specify whether it's internally or externally tangent, just that it's tangent at (1,2). So, perhaps both are valid, but the problem might expect both solutions.Wait, but in the initial steps, I used the condition that the centers are colinear with (1,2), which is correct for tangency. So, both solutions are valid.But in the problem statement, it's just asking for "the equation of the circle", so maybe both are acceptable. But perhaps in the original problem, there's only one solution. Let me check my calculations again.Wait, when I solved for h, I got h = 3.5 and h = -1.5, leading to two centers. So, both circles are valid, one internally tangent and one externally tangent.But the problem says "has a radius of (5√2)/2". So, both circles have that radius, so both are valid.Therefore, the answer should include both equations.But in the initial problem, it's just asking for "the equation", so maybe both are acceptable. Alternatively, perhaps only one is correct based on the position.Wait, let me think about the position of (1,2). It's located at (1,2), which is to the southwest of circle C's center (3,4). So, the externally tangent circle would be on the opposite side, while the internally tangent circle would be on the same side.But since the desired circle has a larger radius, it's possible to have both.But perhaps the problem expects both solutions. So, I should present both equations.Therefore, the equations are:(x - 3.5)² + (y - 4.5)² = 12.5and(x + 1.5)² + (y + 0.5)² = 12.5But to write them in standard form without decimals, I can convert 3.5 to 7/2 and 4.5 to 9/2, and -1.5 to -3/2 and -0.5 to -1/2.So, the equations become:(x - 7/2)² + (y - 9/2)² = (5√2/2)²and(x + 3/2)² + (y + 1/2)² = (5√2/2)²But let me check if both are correct.For the first circle, center (7/2, 9/2):Distance to (1,2):√[(1 - 7/2)² + (2 - 9/2)²] = √[(-5/2)² + (-5/2)²] = √[25/4 + 25/4] = √[50/4] = √12.5 = (5√2)/2, which matches the radius.Distance to circle C's center (3,4):√[(7/2 - 3)² + (9/2 - 4)²] = √[(1/2)² + (1/2)²] = √[1/4 + 1/4] = √(1/2) = √2/2, which is equal to the difference of radii (5√2/2 - 2√2 = √2/2). So, internally tangent.For the second circle, center (-3/2, -1/2):Distance to (1,2):√[(1 + 3/2)² + (2 + 1/2)²] = √[(5/2)² + (5/2)²] = √[25/4 + 25/4] = √[50/4] = √12.5 = (5√2)/2, which matches the radius.Distance to circle C's center (3,4):√[( -3/2 - 3)² + (-1/2 - 4)²] = √[(-9/2)² + (-9/2)²] = √[81/4 + 81/4] = √[162/4] = √40.5 = (9√2)/2, which is equal to the sum of radii (2√2 + 5√2/2 = 9√2/2). So, externally tangent.Therefore, both circles are valid, one internally tangent and one externally tangent at (1,2).But the problem says "the circle", so maybe both are acceptable. Alternatively, perhaps only one is correct based on the position.But since the problem doesn't specify, I think both are correct. So, the equations are:(x - 7/2)² + (y - 9/2)² = (5√2/2)²and(x + 3/2)² + (y + 1/2)² = (5√2/2)²But to write them in the standard form without fractions, I can expand them, but the problem might accept the center-radius form.Alternatively, perhaps only one is correct. Let me think again.Wait, the problem says "the circle that is tangent to the circle C at (1,2) and has a radius of (5√2)/2". So, both circles satisfy this condition, so both are correct.Therefore, the final answer should include both equations.But in the initial problem, it's just asking for "the equation", so maybe both are acceptable. Alternatively, perhaps only one is correct based on the position.Wait, perhaps the problem expects only one solution, so maybe I made a mistake in assuming both are valid.Wait, let me think about the geometry again. If the desired circle is tangent to circle C at (1,2), and has a larger radius, it can be either externally tangent or internally tangent. Both are possible, so both circles are valid.Therefore, the answer should include both equations.But to write them in the standard form, I can present them as:(x - 7/2)² + (y - 9/2)² = (5√2/2)²and(x + 3/2)² + (y + 1/2)² = (5√2/2)²But perhaps the problem expects only one, so maybe I should check which one is correct.Wait, let me think about the direction from (3,4) to (1,2). The vector is (-2, -2), which points towards the southwest. So, the center of the desired circle should lie along this line, either beyond (1,2) or towards (3,4).For the internally tangent circle, the center is (7/2, 9/2) which is (3.5,4.5), which is in the direction opposite to (1,2) from (3,4), so it's towards the northeast, which would mean the circle encloses circle C, but since it's only tangent at (1,2), that's possible.For the externally tangent circle, the center is (-3/2, -1/2), which is in the direction beyond (1,2) from (3,4), so it's towards the southwest, which is the same direction as the point of tangency.Therefore, both are valid, but perhaps the problem expects both solutions.But since the problem says "the circle", maybe only one is expected, but I think both are correct.Therefore, the equations are:(x - 7/2)² + (y - 9/2)² = (5√2/2)²and(x + 3/2)² + (y + 1/2)² = (5√2/2)²But to write them without fractions, I can multiply out:First circle:(x - 7/2)² + (y - 9/2)² = 25/2Expanding:x² -7x + 49/4 + y² -9y + 81/4 = 25/2Combine constants:49/4 + 81/4 = 130/4 = 65/2So:x² + y² -7x -9y + 65/2 = 25/2Subtract 25/2:x² + y² -7x -9y + 40/2 = 0Simplify:x² + y² -7x -9y + 20 = 0Similarly, for the second circle:(x + 3/2)² + (y + 1/2)² = 25/2Expanding:x² +3x + 9/4 + y² + y + 1/4 = 25/2Combine constants:9/4 + 1/4 = 10/4 = 5/2So:x² + y² +3x + y + 5/2 = 25/2Subtract 25/2:x² + y² +3x + y - 20/2 = 0Simplify:x² + y² +3x + y -10 = 0Therefore, the two possible equations are:x² + y² -7x -9y +20 = 0andx² + y² +3x + y -10 = 0But let me check if both are correct.For the first equation, plugging in (1,2):1 +4 -7 -18 +20 = 1+4=5; 5-7=-2; -2-18=-20; -20+20=0. Correct.For the second equation, plugging in (1,2):1 +4 +3 +2 -10 = 1+4=5; 5+3=8; 8+2=10; 10-10=0. Correct.So, both equations are correct.Therefore, the final answer is both equations:x² + y² -7x -9y +20 = 0andx² + y² +3x + y -10 = 0But the problem says "the equation", so maybe both are acceptable. Alternatively, perhaps only one is correct based on the position.But since both are valid, I think both should be presented.So, the equations are:(x - 7/2)² + (y - 9/2)² = (5√2/2)²and(x + 3/2)² + (y + 1/2)² = (5√2/2)²Or in expanded form:x² + y² -7x -9y +20 = 0andx² + y² +3x + y -10 = 0But the problem might expect the standard form, so I'll present both.

💡First, I'll address the second congruence to find the value of ( c ):[7c equiv 3 pmod{8}]To solve for ( c ), I need to find the modular inverse of 7 modulo 8. Since ( 7 times 7 = 49 equiv 1 pmod{8} ), the inverse of 7 is 7 itself. Therefore:[c equiv 7 times 3 equiv 21 equiv 5 pmod{8}]So, ( c = 5 ).Next, I'll solve the third congruence to find ( b ):[5b equiv 4 + b pmod{8}]Subtracting ( b ) from both sides:[4b equiv 4 pmod{8}]Dividing both sides by 4 (noting that 4 and 8 are not coprime, so I'll consider possible values of ( b ) that satisfy this equation):[b equiv 1 pmod{8}]Thus, ( b = 1 ).Now, substituting ( b = 1 ) and ( c = 5 ) into the first congruence:[a times 1 times 5 equiv 1 pmod{8}]Simplifying:[5a equiv 1 pmod{8}]To solve for ( a ), I need the modular inverse of 5 modulo 8. Since ( 5 times 5 = 25 equiv 1 pmod{8} ), the inverse of 5 is 5 itself. Therefore:[a equiv 5 pmod{8}]So, ( a = 5 ).Finally, I'll calculate ( a + b + c ) and find the remainder when divided by 8:[a + b + c = 5 + 1 + 5 = 11][11 div 8 = 1 text{ with a remainder of } 3]Therefore, the remainder when ( a + b + c ) is divided by 8 is 3.