💡First, identify the two largest one-digit primes, which are 5 and 7.Next, determine the largest two-digit prime, which is 97.Then, find the largest three-digit prime, which is 997.Now, calculate the product of these primes: 5 × 7 × 97 × 997.First, multiply 5 and 7 to get 35.Next, multiply 35 by 97 to obtain 3395.Finally, multiply 3395 by 997. To simplify, recognize that 997 is 1000 minus 3.So, 3395 × 997 = 3395 × (1000 - 3) = 3395000 - 10185 = 3383815.Thus, the final product is 3383815.

💡First, I need to simplify the angle (1030^{circ}) to find an equivalent angle within the standard range of (0^{circ}) to (360^{circ}). To do this, I'll subtract multiples of (360^{circ}) until the angle is within the desired range.Starting with (1030^{circ}), I subtract (360^{circ}) three times:[1030^{circ} - 3 times 360^{circ} = 1030^{circ} - 1080^{circ} = -50^{circ}]So, (1030^{circ}) is equivalent to (-50^{circ}).Next, I'll use the property of the cosine function that states:[cos(-theta) = cos(theta)]Applying this property to (-50^{circ}):[cos(-50^{circ}) = cos(50^{circ})]Therefore, (cos(1030^{circ}) = cos(50^{circ})).Looking at the options provided:- A: (cos 50^{circ})- B: (-cos 50^{circ})- C: (sin 50^{circ})- D: (-sin 50^{circ})The correct answer is option A.

💡First, I need to determine the total area occupied by the bedrooms and bathrooms in Raj's house. Each bedroom measures 11 feet by 11 feet, so the area of one bedroom is:11 feet * 11 feet = 121 square feetWith 4 bedrooms, the total area for all bedrooms is:121 square feet/bedroom * 4 bedrooms = 484 square feetEach bathroom measures 6 feet by 8 feet, so the area of one bathroom is:6 feet * 8 feet = 48 square feetWith 2 bathrooms, the total area for both bathrooms is:48 square feet/bathroom * 2 bathrooms = 96 square feetAdding the areas of the bedrooms and bathrooms gives:484 square feet (bedrooms) + 96 square feet (bathrooms) = 580 square feetThe total area of Raj's house is 1,110 square feet. Subtracting the area occupied by bedrooms and bathrooms from the total area leaves:1,110 square feet (total area) - 580 square feet (bedrooms and bathrooms) = 530 square feetSince the kitchen and living area are of equal size, I divide the remaining area by 2 to find the area of the kitchen:530 square feet / 2 = 265 square feetTherefore, the area of the kitchen is 265 square feet.

💡Alright, so I've got this problem here about functions and translations, and I need to find the minimum value of |2x₁ + x₂| given some conditions. Let me try to break it down step by step.First, the function f(x) is given as 2 sin x cos x. Hmm, I remember that 2 sin x cos x is actually equal to sin 2x. Yeah, that's one of those double-angle identities. So, f(x) simplifies to sin 2x. Cool, that makes it a bit easier to work with.Next, the graph of f(x) is translated π/12 units to the left. Translating a function to the left by π/12 means replacing x with (x + π/12) in the function. So, the new function after the translation would be sin[2(x + π/12)]. Let me compute that: 2(x + π/12) is 2x + π/6. So, the translated function is sin(2x + π/6). Then, the graph is translated 1 unit upward. Translating upward by 1 unit means adding 1 to the entire function. So, the final function g(x) is sin(2x + π/6) + 1. Got that down.Now, the problem states that f(x₁)g(x₂) = 2. So, I need to find x₁ and x₂ such that when I multiply f(x₁) and g(x₂), I get 2. Then, I have to find the minimum value of |2x₁ + x₂|.Let me write down what I have:f(x) = sin 2xg(x) = sin(2x + π/6) + 1So, f(x₁) = sin(2x₁)g(x₂) = sin(2x₂ + π/6) + 1And f(x₁) * g(x₂) = 2So, sin(2x₁) * [sin(2x₂ + π/6) + 1] = 2Hmm, okay. Let me think about the maximum and minimum values of sine functions because that might help. The sine function ranges between -1 and 1, so sin(2x₁) can be at most 1 and at least -1. Similarly, sin(2x₂ + π/6) can be between -1 and 1, so sin(2x₂ + π/6) + 1 can be between 0 and 2.Given that f(x₁)*g(x₂) = 2, and since the maximum value of f(x₁) is 1 and the maximum value of g(x₂) is 2, their product can be at most 2. So, the maximum possible product is 2, which occurs when f(x₁) = 1 and g(x₂) = 2.Therefore, to satisfy f(x₁)*g(x₂) = 2, we must have f(x₁) = 1 and g(x₂) = 2. That makes sense because if either one is less than their maximum, the product would be less than 2.So, let's set up the equations:1. sin(2x₁) = 12. sin(2x₂ + π/6) + 1 = 2Let's solve the first equation: sin(2x₁) = 1When does sin θ = 1? That's at θ = π/2 + 2πk, where k is any integer. So, 2x₁ = π/2 + 2πkTherefore, x₁ = π/4 + πkOkay, so x₁ can be π/4, 5π/4, 9π/4, etc., depending on the integer k.Now, the second equation: sin(2x₂ + π/6) + 1 = 2Subtract 1 from both sides: sin(2x₂ + π/6) = 1Again, when does sin θ = 1? At θ = π/2 + 2πn, where n is any integer.So, 2x₂ + π/6 = π/2 + 2πnLet's solve for x₂:2x₂ = π/2 - π/6 + 2πnCompute π/2 - π/6: π/2 is 3π/6, so 3π/6 - π/6 = 2π/6 = π/3So, 2x₂ = π/3 + 2πnTherefore, x₂ = π/6 + πnSo, x₂ can be π/6, 7π/6, 13π/6, etc., depending on the integer n.Alright, so now we have expressions for x₁ and x₂ in terms of integers k and n.x₁ = π/4 + πkx₂ = π/6 + πnWe need to find the minimum value of |2x₁ + x₂|Let's substitute the expressions for x₁ and x₂ into 2x₁ + x₂:2x₁ + x₂ = 2*(π/4 + πk) + (π/6 + πn)Compute 2*(π/4): that's π/2So, 2x₁ + x₂ = π/2 + 2πk + π/6 + πnCombine the constants: π/2 + π/6π/2 is 3π/6, so 3π/6 + π/6 = 4π/6 = 2π/3So, 2x₁ + x₂ = 2π/3 + 2πk + πnTherefore, |2x₁ + x₂| = |2π/3 + 2πk + πn|We need to find the minimum value of this expression over integers k and n.So, essentially, we're looking for the smallest absolute value of 2π/3 + 2πk + πn.Let me factor out π:|2π/3 + π(2k + n)| = π|2/3 + 2k + n|So, we need to minimize |2/3 + 2k + n|, and then multiply by π.Let me denote m = 2k + n, where m is an integer because k and n are integers.So, the expression becomes |2/3 + m|, and we need to find the integer m that makes this as small as possible.What integer m minimizes |2/3 + m|?Well, 2/3 is approximately 0.6667. So, the closest integers are m = 0 and m = -1.Compute |2/3 + 0| = 2/3 ≈ 0.6667Compute |2/3 + (-1)| = |-1/3| = 1/3 ≈ 0.3333So, m = -1 gives a smaller value.Therefore, the minimum value of |2/3 + m| is 1/3.Hence, the minimum value of |2x₁ + x₂| is π*(1/3) = π/3.So, the answer is π/3, which is option B.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Simplified f(x) correctly to sin 2x.2. Translated left by π/12: replaced x with x + π/12, so 2(x + π/12) = 2x + π/6. Correct.3. Translated up by 1: added 1 to the function. So, g(x) = sin(2x + π/6) + 1. Correct.4. Set f(x₁)g(x₂) = 2. Since f(x) can be at most 1 and g(x) can be at most 2, their product is at most 2. So, to get 2, f(x₁) must be 1 and g(x₂) must be 2. That makes sense.5. Solved sin(2x₁) = 1: 2x₁ = π/2 + 2πk, so x₁ = π/4 + πk. Correct.6. Solved sin(2x₂ + π/6) + 1 = 2: sin(2x₂ + π/6) = 1, so 2x₂ + π/6 = π/2 + 2πn, leading to x₂ = π/6 + πn. Correct.7. Plugged into 2x₁ + x₂: 2*(π/4 + πk) + (π/6 + πn) = π/2 + 2πk + π/6 + πn = 2π/3 + π(2k + n). Correct.8. Expressed as π|2/3 + m| where m is integer. Correct.9. Found m = -1 gives the smallest |2/3 -1| = 1/3. So, minimum |2x₁ + x₂| is π/3. Correct.I think all the steps are solid. So, yeah, the minimum value is π/3.**Final Answer**boxed{B}

💡To solve the problem, let's denote the two positive integers as ( a ) and ( b ) with ( a > b ). The given equation is:[ab + a + b = 71]Using Simon's Favorite Factoring Trick, we add 1 to both sides to facilitate factoring:[ab + a + b + 1 = 72]This can be factored as:[(a + 1)(b + 1) = 72]Next, we list the factor pairs of 72:- ( (72, 1) ): Results in ( a = 71 ) and ( b = 0 ), which are out of range.- ( (36, 2) ): Results in ( a = 35 ) and ( b = 1 ), which are out of range.- ( (24, 3) ): Results in ( a = 23 ) and ( b = 2 ), which are out of range.- ( (18, 4) ): Results in ( a = 17 ) and ( b = 3 ), which are out of range.- ( (12, 6) ): Results in ( a = 11 ) and ( b = 5 ), which are within range and relatively prime.Thus, the valid pair is ( a = 11 ) and ( b = 5 ). Their sum is:[a + b = 11 + 5 = 16]

💡Okay, so I have this problem about a square ABCD with side length 2. Point E is the midpoint of side CD, and I need to find the dot product of vectors AE and BD. Hmm, let me try to visualize this square first. Let me draw it out in my mind: square ABCD, so the vertices are A, B, C, D in order, forming a square. Each side is length 2. E is the midpoint of CD, so that means E divides CD into two equal parts, each of length 1.Now, I need to find the vectors AE and BD. Let me recall that vectors can be represented in terms of their coordinates if I assign a coordinate system to the square. Maybe it's easier if I place the square on a coordinate plane. Let me choose point A as the origin (0,0). Then, since it's a square with side length 2, point B would be at (2,0), point C at (2,2), and point D at (0,2). Wait, hold on, if E is the midpoint of CD, then CD goes from (2,2) to (0,2). The midpoint E would be halfway between these two points. So, the coordinates of E would be the average of the x-coordinates and the average of the y-coordinates of C and D. So, E's x-coordinate is (2 + 0)/2 = 1, and y-coordinate is (2 + 2)/2 = 2. So, E is at (1,2).Alright, now I need to find vectors AE and BD. Vector AE goes from point A to point E. Since A is at (0,0) and E is at (1,2), vector AE can be represented as (1,2). Similarly, vector BD goes from point B to point D. Point B is at (2,0) and point D is at (0,2), so vector BD is D - B, which is (0 - 2, 2 - 0) = (-2,2).Now, I need to compute the dot product of vectors AE and BD. The dot product of two vectors (a,b) and (c,d) is given by ac + bd. So, applying that here, vector AE is (1,2) and vector BD is (-2,2). So, their dot product is (1)(-2) + (2)(2) = -2 + 4 = 2.Wait, that seems straightforward. Let me double-check my steps to make sure I didn't make any mistakes. First, assigning coordinates: A(0,0), B(2,0), C(2,2), D(0,2). E is the midpoint of CD, so E is at (1,2). Vector AE is from A(0,0) to E(1,2), which is (1,2). Vector BD is from B(2,0) to D(0,2), which is (-2,2). Dot product is (1)(-2) + (2)(2) = -2 + 4 = 2. Yep, that seems correct.Alternatively, maybe I can solve this without coordinates, just using vector properties. Let me think. In a square, the diagonals are equal and bisect each other at 90 degrees. Wait, no, in a square, the diagonals bisect each other at 90 degrees only if it's a square, which it is. So, BD is a diagonal, and AE is not a diagonal but a vector from A to the midpoint of CD.Hmm, maybe I can express vectors AE and BD in terms of the sides of the square. Let me denote vector AB as vector i and vector AD as vector j. Since it's a square, these vectors are perpendicular and have the same magnitude, which is 2.So, vector AB = (2,0) and vector AD = (0,2). Then, vector BD is vector AD - vector AB, which is (0,2) - (2,0) = (-2,2). Vector AE goes from A to E, which is the midpoint of CD. Since CD is from C(2,2) to D(0,2), the midpoint E is at (1,2). So, vector AE is from A(0,0) to E(1,2), which is (1,2).Expressed in terms of vectors AB and AD, vector AE can be written as vector AD + (1/2) vector AB. Because from A, moving halfway along AB (which is vector AB scaled by 1/2) and then moving up to E, which is vector AD. So, vector AE = (1/2) vector AB + vector AD.So, vector AE = (1/2)i + j, and vector BD = j - i. Then, the dot product of AE and BD is [(1/2)i + j] · [j - i]. Let's compute this:(1/2)i · j - (1/2)i · i + j · j - j · i.Now, since i and j are perpendicular, their dot product is zero. So, i · j = 0. Also, i · i = |i|² = 4, and j · j = |j|² = 4. So, substituting these in:(1/2)(0) - (1/2)(4) + 4 - 0 = 0 - 2 + 4 = 2.So, again, the dot product is 2. That confirms my earlier result.Wait, just to make sure, maybe I can use another method, like coordinate geometry, to find the vectors and their dot product. Let me assign coordinates again, but this time, perhaps place the square differently. Maybe place point A at (0,0), B at (0,2), C at (-2,2), and D at (-2,0). Wait, no, that would change the orientation, but the side lengths are still 2. Hmm, but E is the midpoint of CD, so CD would be from (-2,2) to (-2,0), so midpoint E would be at (-2,1). Then vector AE would be from A(0,0) to E(-2,1), which is (-2,1). Vector BD is from B(0,2) to D(-2,0), which is (-2,-2). Then, the dot product is (-2)(-2) + (1)(-2) = 4 - 2 = 2. Again, same result. So, regardless of how I place the square on the coordinate system, the dot product comes out to 2.Alternatively, maybe I can use vector projections. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. So, |AE| * |BD| * cos(theta) = ?First, let's compute |AE|. Vector AE is (1,2), so its magnitude is sqrt(1² + 2²) = sqrt(1 + 4) = sqrt(5). Vector BD is (-2,2), so its magnitude is sqrt((-2)² + 2²) = sqrt(4 + 4) = sqrt(8) = 2*sqrt(2). The angle between them can be found using the dot product formula: cos(theta) = (AE · BD) / (|AE| |BD|). But wait, that's circular because we're trying to find AE · BD. So, maybe this approach isn't helpful unless we already know the angle.Alternatively, maybe I can find the angle between AE and BD using geometry. Let me see. In the square, BD is a diagonal, and AE is a vector from A to the midpoint of CD. Maybe I can find the angle between them using triangle properties or something.But honestly, the coordinate method seems straightforward and reliable here. I think I've confirmed it enough times. So, the dot product is 2.Wait, just to be thorough, let me compute the vectors again. Vector AE is from A(0,0) to E(1,2), so it's (1,2). Vector BD is from B(2,0) to D(0,2), so it's (-2,2). Dot product is (1)(-2) + (2)(2) = -2 + 4 = 2. Yep, that's consistent.I think I'm confident that the answer is 2.