💡Okay, so I have this problem where I need to find how many four-digit numbers, represented as (overline{abcd}), are divisible by 3. The catch is that the first three digits (a), (b), and (c) are permutations of three consecutive integers. Hmm, that sounds a bit tricky, but let me break it down step by step.First, let me recall that a number is divisible by 3 if the sum of its digits is divisible by 3. So, for (overline{abcd}) to be divisible by 3, (a + b + c + d) must be a multiple of 3. That's a key point.Now, (a), (b), and (c) are permutations of three consecutive integers. That means they could be something like 1, 2, 3 or 5, 6, 7, but in any order. So, I need to figure out all possible sets of three consecutive integers that can be arranged in any order for the first three digits.Let me denote the three consecutive integers as (k), (k+1), and (k+2), where (k) is an integer. Since (a), (b), and (c) are digits, they must each be between 0 and 9. However, (a) can't be 0 because it's the first digit of a four-digit number. So, (k) must be such that (k), (k+1), and (k+2) are all valid digits, and (a) is non-zero.Let me list the possible values of (k):- If (k = 0), the digits are 0, 1, 2. But since (a) can't be 0, we have to exclude permutations where the first digit is 0. So, how many valid permutations are there for (k=0)?Wait, for (k=0), the digits are 0, 1, 2. The total permutations are 6, but some of them start with 0, which isn't allowed. How many start with 0? Well, if 0 is fixed as the first digit, the remaining two digits can be arranged in 2 ways: 0,1,2 and 0,2,1. So, there are 2 invalid permutations. Therefore, for (k=0), we have 6 - 2 = 4 valid permutations.Okay, moving on. For (k=1), the digits are 1, 2, 3. All permutations are valid because none of them start with 0. There are 6 permutations here.Similarly, for (k=2), digits are 2, 3, 4. Again, 6 permutations, all valid.Continuing this way, (k) can go up to 7 because if (k=7), the digits are 7, 8, 9, which are all valid. If (k=8), the digits would be 8, 9, 10, but 10 isn't a single digit, so that's invalid. So, (k) can be from 0 to 7.So, for each (k) from 1 to 7, we have 6 valid permutations, and for (k=0), we have 4 valid permutations.Let me calculate the total number of valid ((a, b, c)) triplets:- For (k=0): 4 permutations- For (k=1) to (k=7): 7 values, each with 6 permutations, so 7 * 6 = 42Total permutations: 4 + 42 = 46Okay, so there are 46 possible ways to arrange (a), (b), and (c) such that they are permutations of three consecutive integers, with (a neq 0).Now, for each of these 46 triplets, we need to find the number of possible (d) digits such that the entire number (overline{abcd}) is divisible by 3. Remember, (d) is a single digit (0-9), and the sum (a + b + c + d) must be divisible by 3.Let me denote the sum of (a + b + c) as (S). Then, for each triplet, (S) is fixed, and we need (S + d) to be divisible by 3. So, (d) must be congruent to (-S mod 3). Since (d) is a digit, it must be one of 0, 1, 2, ..., 9.But how many choices of (d) satisfy this condition? Well, for any given (S), the possible (d) values are those that make (S + d) divisible by 3. Since (d) can be 0-9, there are 10 possible digits, and exactly one-third of them (rounded appropriately) will satisfy the condition.But actually, since 10 divided by 3 is approximately 3.333, but since 10 isn't a multiple of 3, the number of valid (d) digits depends on the value of (S mod 3).Wait, let me think. For any given (S), (d) must be congruent to (-S mod 3). So, (d) can be any digit such that (d equiv (-S) mod 3). The digits 0-9 cover all residues modulo 3:- Digits congruent to 0 mod 3: 0, 3, 6, 9 (4 digits)- Digits congruent to 1 mod 3: 1, 4, 7 (3 digits)- Digits congruent to 2 mod 3: 2, 5, 8 (3 digits)So, depending on what (-S mod 3) is, (d) can be one of these sets. If (-S equiv 0 mod 3), then (d) can be 0, 3, 6, or 9 (4 choices). If (-S equiv 1 mod 3), then (d) can be 1, 4, or 7 (3 choices). Similarly, if (-S equiv 2 mod 3), (d) can be 2, 5, or 8 (3 choices).Therefore, for each triplet, the number of valid (d) digits is either 3 or 4, depending on (S mod 3).But wait, (S = a + b + c = k + (k+1) + (k+2) = 3k + 3). So, (S = 3(k + 1)). Therefore, (S) is always a multiple of 3. So, (S equiv 0 mod 3). Therefore, (-S equiv 0 mod 3). So, (d) must be congruent to 0 mod 3. Therefore, (d) can be 0, 3, 6, or 9. That's 4 choices.Wait, that's a key insight! Since (S = 3(k + 1)), which is divisible by 3, (d) must also be divisible by 3 to keep the total sum divisible by 3. So, (d) can be 0, 3, 6, or 9. That's 4 choices for (d), regardless of the value of (k).So, for each of the 46 triplets, there are 4 possible (d) digits. Therefore, the total number of four-digit numbers is 46 * 4 = 184.Wait, hold on. Let me double-check that. For each triplet, regardless of (k), (S) is a multiple of 3, so (d) must be 0, 3, 6, or 9. That gives 4 choices for (d). So, 46 triplets * 4 digits = 184 numbers.But let me make sure I didn't make a mistake earlier. When (k=0), the digits are 0,1,2, but (a) can't be 0. So, the valid permutations are 4, as I calculated. For each of these 4 triplets, (d) can be 0,3,6,9, so 4 * 4 = 16 numbers.For (k=1) to (k=7), each has 6 triplets, so 7 * 6 = 42 triplets. Each of these 42 triplets can have 4 choices for (d), so 42 * 4 = 168 numbers.Adding them together: 16 + 168 = 184.Yes, that seems consistent.But wait, hold on a second. When (k=0), the digits are 0,1,2, but (d) can be 0,3,6,9. However, (d) is the last digit, so it can be 0. So, including 0 is fine because it's the last digit, not the first. So, 4 choices for (d) are valid, even if (d=0). So, that part is okay.Another thing to check: when (k=7), the digits are 7,8,9. So, all are non-zero, so all 6 permutations are valid, and (d) can be 0,3,6,9. So, that's 6 * 4 = 24 numbers for (k=7).Similarly, for other (k) values, it's consistent.Therefore, I think my calculation is correct: 46 triplets * 4 digits = 184 four-digit numbers.But just to be thorough, let me think about whether there's any overlap or overcounting.Each triplet is unique because (k) ranges from 0 to 7, and for each (k), the triplets are distinct. So, no overlaps there. And for each triplet, the digit (d) is independent, so no overcounting in that aspect either.Therefore, I feel confident that the total number of such four-digit numbers is 184.**Final Answer**The total number of such four-digit numbers is (boxed{184}).

💡Okay, so I have this problem where I'm given two expressions for m and n in terms of b. The problem is asking me to figure out the relationship between m and n. The options are whether m is greater than n, greater than or equal to n, less than n, or less than or equal to n. First, let me write down the given equations to make sure I have them right:m = 2b + 2022n = b² + 2023Alright, so m is a linear function of b, and n is a quadratic function of b. I need to compare these two expressions to see how m and n relate.Maybe I can start by trying to express the relationship between m and n. If I subtract n from m, I can see whether the result is positive, negative, or zero. That should tell me whether m is greater than, less than, or equal to n.So, let's compute m - n:m - n = (2b + 2022) - (b² + 2023)Let me simplify that step by step. First, distribute the negative sign to both terms in n:= 2b + 2022 - b² - 2023Now, combine like terms. The constants 2022 and -2023 can be combined:2022 - 2023 = -1So now we have:= -b² + 2b - 1Hmm, that's a quadratic expression. Maybe I can factor it to see if it simplifies further. Let's see:Looking at -b² + 2b - 1, I notice that all the coefficients are negative except for the 2b term. Perhaps I can factor out a negative sign to make it easier:= -(b² - 2b + 1)Now, inside the parentheses, b² - 2b + 1 is a perfect square. It factors into (b - 1)².So, substituting that back in:= -(b - 1)²Okay, so m - n simplifies to -(b - 1)². Now, since (b - 1)² is always non-negative (because any real number squared is non-negative), multiplying it by -1 makes it non-positive. That means -(b - 1)² is always less than or equal to zero.So, m - n ≤ 0, which implies that m ≤ n.Wait, but let me double-check that. If I have m - n = -(b - 1)², then:- If b = 1, then m - n = -(1 - 1)² = -0 = 0, so m = n.- If b ≠ 1, then (b - 1)² is positive, so m - n is negative, meaning m < n.Therefore, in all cases, m is either less than or equal to n. So, the correct relationship is m ≤ n.Let me just test this with some values of b to make sure.For example, let's take b = 0:m = 2(0) + 2022 = 2022n = (0)² + 2023 = 2023So, m = 2022 and n = 2023, which means m < n.Another example, b = 1:m = 2(1) + 2022 = 2 + 2022 = 2024n = (1)² + 2023 = 1 + 2023 = 2024So, m = n.And one more, b = 2:m = 2(2) + 2022 = 4 + 2022 = 2026n = (2)² + 2023 = 4 + 2023 = 2027Again, m = 2026 < n = 2027.What if b is negative? Let's try b = -1:m = 2(-1) + 2022 = -2 + 2022 = 2020n = (-1)² + 2023 = 1 + 2023 = 2024So, m = 2020 < n = 2024.It seems consistent across different values of b. So, my conclusion is that m is always less than or equal to n, with equality only when b = 1.Therefore, the correct answer should be D: m ≤ n.

💡Okay, so I have this problem where I'm given that the tangent of theta is 3, and I need to find the tangent of three theta. Hmm, I remember there are some trigonometric identities that relate tangent of multiple angles to the tangent of a single angle. Maybe it's the triple angle formula? I think that's what it's called.Let me try to recall the formula for tan(3θ). I think it involves tanθ in some way, maybe with some coefficients. I believe it's something like (3 tanθ - tan³θ) divided by (1 - 3 tan²θ). Wait, is that right? Let me check. If I remember correctly, the triple angle formula for tangent is indeed:tan(3θ) = (3 tanθ - tan³θ) / (1 - 3 tan²θ)Okay, so that seems correct. Now, since I know that tanθ is 3, I can substitute that value into the formula. Let me write that down step by step.First, substitute tanθ = 3 into the numerator:3 tanθ = 3 * 3 = 9tan³θ = (3)³ = 27So, the numerator becomes 9 - 27, which is -18.Now, let's work on the denominator. Again, substitute tanθ = 3:3 tan²θ = 3 * (3)² = 3 * 9 = 27So, the denominator is 1 - 27, which is -26.Putting it all together, tan(3θ) = (-18) / (-26). Hmm, both numerator and denominator are negative, so they cancel out, giving me 18/26.Wait, 18 and 26 have a common factor. Let me simplify that fraction. Both are divisible by 2:18 ÷ 2 = 926 ÷ 2 = 13So, the simplified fraction is 9/13.Therefore, tan(3θ) is 9/13 when tanθ is 3.But just to make sure I didn't make any mistakes, let me verify each step again.1. I used the correct triple angle formula for tangent. Yes, I think that's right because I remember it's similar to the double angle formula but with higher powers.2. Substituted tanθ = 3 correctly into the formula. 3*3 is 9, and 3³ is 27. So, numerator is 9 - 27 = -18. Denominator is 1 - 3*(3²) = 1 - 27 = -26.3. Divided -18 by -26, which gives positive 18/26, which simplifies to 9/13. That seems correct.I don't see any errors in my calculations. So, I think the answer is 9/13.Wait, just to be thorough, maybe I should consider if there's another method to solve this, like using sine and cosine. Let me try that approach to cross-verify.If tanθ = 3, then I can think of a right triangle where the opposite side is 3 and the adjacent side is 1, so the hypotenuse would be sqrt(1² + 3²) = sqrt(10). Therefore, sinθ = 3/sqrt(10) and cosθ = 1/sqrt(10).Now, using the triple angle formula for sine and cosine:sin(3θ) = 3 sinθ - 4 sin³θcos(3θ) = 4 cos³θ - 3 cosθThen, tan(3θ) = sin(3θ)/cos(3θ)Let me compute sin(3θ):sin(3θ) = 3*(3/sqrt(10)) - 4*(3/sqrt(10))³First term: 3*(3/sqrt(10)) = 9/sqrt(10)Second term: 4*(27)/(sqrt(10))³ = 4*27/(10*sqrt(10)) = 108/(10*sqrt(10)) = 54/(5*sqrt(10))So, sin(3θ) = 9/sqrt(10) - 54/(5*sqrt(10)) = (45/5 - 54/5)/sqrt(10) = (-9/5)/sqrt(10) = -9/(5*sqrt(10))Now, compute cos(3θ):cos(3θ) = 4*(1/sqrt(10))³ - 3*(1/sqrt(10))First term: 4*(1)/(10*sqrt(10)) = 4/(10*sqrt(10)) = 2/(5*sqrt(10))Second term: 3*(1/sqrt(10)) = 3/sqrt(10)So, cos(3θ) = 2/(5*sqrt(10)) - 3/sqrt(10) = (2/5 - 3)/sqrt(10) = (-13/5)/sqrt(10) = -13/(5*sqrt(10))Now, tan(3θ) = sin(3θ)/cos(3θ) = (-9/(5*sqrt(10))) / (-13/(5*sqrt(10))) = (9/5)/(13/5) = 9/13Okay, so using the sine and cosine approach also gives me tan(3θ) = 9/13. That confirms my earlier result.I think I'm confident now that the answer is 9/13.

💡Alright, so I've got this problem here where I need to compare three different values: a, b, and c. The options are given, and I need to figure out which one is correct. Let me take it step by step.First, let's write down what each of these variables is:- ( a = 2^{frac{1}{3}} )- ( b = log_{3} frac{2}{3} )- ( c = log_{frac{1}{2}} frac{1}{3} )I need to compare a, b, and c and determine their order from largest to smallest or vice versa. The options are:A: ( a > b > c )B: ( a > c > b )C: ( c > a > b )D: ( c > b > a )Okay, so let's tackle each one by one.Starting with ( a = 2^{frac{1}{3}} ). That's the cube root of 2. I know that ( 2^{frac{1}{3}} ) is approximately 1.26 because ( 1.26^3 ) is roughly 2. So, a is about 1.26. That's more than 1 but less than 2. Specifically, since ( 2^{frac{1}{2}} ) is about 1.414, which is the square root of 2, the cube root should be a bit smaller, so 1.26 seems right.Next, ( b = log_{3} frac{2}{3} ). Hmm, logarithms can be tricky, especially with bases other than 10 or e. Let me recall that ( log_{a} b = frac{ln b}{ln a} ). So, I can rewrite this as ( frac{ln frac{2}{3}}{ln 3} ). Calculating that, ( ln frac{2}{3} ) is ( ln 2 - ln 3 ), which is approximately 0.6931 - 1.0986 = -0.4055. Then, ( ln 3 ) is about 1.0986. So, ( b ) is approximately ( -0.4055 / 1.0986 approx -0.369 ). So, b is negative.Now, ( c = log_{frac{1}{2}} frac{1}{3} ). Again, using the change of base formula, this is ( frac{ln frac{1}{3}}{ln frac{1}{2}} ). Calculating the numerator: ( ln frac{1}{3} = -ln 3 approx -1.0986 ). The denominator: ( ln frac{1}{2} = -ln 2 approx -0.6931 ). So, ( c ) is ( (-1.0986) / (-0.6931) approx 1.58496 ). So, c is approximately 1.585.Let me summarize:- ( a approx 1.26 )- ( b approx -0.369 )- ( c approx 1.585 )So, putting them in order from largest to smallest:c (1.585) > a (1.26) > b (-0.369)Which corresponds to option C: ( c > a > b ).Wait, let me double-check my calculations to make sure I didn't make any mistakes.For a: ( 2^{1/3} ) is indeed approximately 1.26. That seems correct.For b: ( log_{3} frac{2}{3} ). Since ( frac{2}{3} ) is less than 1, and the base 3 is greater than 1, the logarithm should be negative. My calculation gave me approximately -0.369, which seems reasonable.For c: ( log_{frac{1}{2}} frac{1}{3} ). Since the base is less than 1, the logarithm function is decreasing. So, as the argument ( frac{1}{3} ) is less than 1, the logarithm should be positive. My calculation gave me approximately 1.585, which is more than 1, so that seems correct.Therefore, the order is indeed c > a > b, which is option C.I think that's solid. I don't see any errors in my reasoning or calculations.**Final Answer**boxed{C}

💡Okay, so I need to find the equation of the tangent line to the curve y = f(x) at the point (1, f(1)), where f(x) is given as x minus 4 times the natural logarithm of x. Hmm, let me break this down step by step.First, I remember that to find the equation of a tangent line, I need two things: the slope of the tangent at the specific point and a point through which the tangent passes. In this case, the point is given as (1, f(1)), so I need to find f(1) first.Let me calculate f(1). The function is f(x) = x - 4 ln x. Plugging in x = 1, I get f(1) = 1 - 4 ln(1). Wait, what's ln(1)? I recall that the natural logarithm of 1 is 0 because e^0 = 1. So, f(1) = 1 - 4*0 = 1. Okay, so the point is (1, 1).Now, I need the slope of the tangent line at this point. The slope of the tangent line to a curve at a particular point is given by the derivative of the function evaluated at that point. So, I need to find f'(x), the derivative of f(x), and then plug in x = 1.Let's find f'(x). The function is f(x) = x - 4 ln x. The derivative of x with respect to x is 1, and the derivative of ln x is 1/x. So, applying the derivative rules, f'(x) should be 1 - 4*(1/x). Simplifying that, f'(x) = 1 - 4/x.Now, evaluate f'(x) at x = 1. Plugging in, f'(1) = 1 - 4/1 = 1 - 4 = -3. So, the slope of the tangent line at x = 1 is -3.Alright, now I have the slope and a point on the tangent line. To write the equation of the tangent line, I can use the point-slope form, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point.Substituting the values I have, m = -3, x1 = 1, and y1 = 1. So, the equation becomes y - 1 = -3(x - 1). Let me simplify this.Expanding the right side: y - 1 = -3x + 3. Now, adding 1 to both sides to solve for y: y = -3x + 3 + 1, which simplifies to y = -3x + 4.Alternatively, I can write this in standard form, which is Ax + By + C = 0. Starting from y = -3x + 4, I can add 3x to both sides: 3x + y = 4. Then, subtracting 4 from both sides: 3x + y - 4 = 0.So, the equation of the tangent line is 3x + y - 4 = 0. Let me double-check my calculations to make sure I didn't make any mistakes.First, f(1) = 1 - 4 ln(1) = 1 - 0 = 1. That seems correct. Then, f'(x) = 1 - 4/x, so f'(1) = 1 - 4 = -3. That also looks right. Using point-slope form: y - 1 = -3(x - 1). Expanding gives y = -3x + 4, which in standard form is 3x + y - 4 = 0. Yep, everything checks out.I think that's the correct equation for the tangent line at the point (1, 1) on the curve y = f(x).

💡Okay, so I have this problem here: If 8a plus 3b equals zero, and a is three less than b, what is 9b? Hmm, let me try to figure this out step by step. First, I need to understand what the problem is asking. There are two equations given. The first one is 8a + 3b = 0, which is a linear equation involving variables a and b. The second piece of information is that a is three less than b. That means a is equal to b minus 3. So, I can write that as a = b - 3. Alright, now I have two equations:1. 8a + 3b = 02. a = b - 3Since I have two equations and two variables, I can solve this system of equations to find the values of a and b. Once I find b, I can easily compute 9b, which is what the problem is asking for.Let me write down the equations again for clarity:1. 8a + 3b = 02. a = b - 3Now, since the second equation already expresses a in terms of b, I can substitute this expression into the first equation to solve for b. This method is called substitution, and it's a common technique for solving systems of equations.So, substituting a = b - 3 into the first equation:8(b - 3) + 3b = 0Let me simplify this step by step. First, I'll distribute the 8 into the parentheses:8 * b = 8b8 * (-3) = -24So, the equation becomes:8b - 24 + 3b = 0Now, I'll combine like terms. The terms with b are 8b and 3b, so adding them together:8b + 3b = 11bSo, the equation simplifies to:11b - 24 = 0Now, I need to solve for b. I'll isolate b by first adding 24 to both sides of the equation:11b - 24 + 24 = 0 + 2411b = 24Next, I'll divide both sides by 11 to solve for b:11b / 11 = 24 / 11b = 24/11Okay, so b is equal to 24/11. Now, the problem asks for 9b, so I need to multiply b by 9:9b = 9 * (24/11)Let me compute that. First, multiply 9 by 24:9 * 24 = 216So, 9b = 216/11Hmm, 216 divided by 11. Let me check if this can be simplified further. 11 goes into 216 how many times? 11 * 19 = 209, and 216 - 209 = 7. So, 216/11 is equal to 19 and 7/11, but since the problem doesn't specify the form, and 216/11 is already in its simplest fractional form, I think that's the answer.Just to make sure I didn't make any mistakes, let me go back through my steps.1. I started with the two equations: 8a + 3b = 0 and a = b - 3.2. Substituted a in the first equation with b - 3, resulting in 8(b - 3) + 3b = 0.3. Distributed the 8: 8b - 24 + 3b = 0.4. Combined like terms: 11b - 24 = 0.5. Solved for b: 11b = 24 => b = 24/11.6. Calculated 9b: 9 * (24/11) = 216/11.Everything seems to check out. I don't see any errors in my calculations. So, I'm confident that 9b is equal to 216/11.Just to double-check, I can plug the value of b back into the original equations to ensure consistency.First, let's find a using a = b - 3:a = (24/11) - 3 = (24/11) - (33/11) = (-9/11)Now, plug a and b into the first equation:8a + 3b = 8*(-9/11) + 3*(24/11) = (-72/11) + (72/11) = 0Yes, that works out. The left side equals the right side, which is 0. So, the solution is consistent.Therefore, my final answer is 216/11.