💡Alright, so I have this problem here about sets and complements. Let me try to understand what it's asking. The universal set U is defined as all natural numbers less than 8. Okay, so U = {x | x < 8, x ∈ N}. That means U includes all natural numbers starting from 0 up to 7, right? So U should be {0, 1, 2, 3, 4, 5, 6, 7}. Got that down.Now, set A is given by A = {x | (x - 1)(x - 3)(x - 4)(x - 7) = 0}. Hmm, that looks like a product of factors equal to zero. I remember from algebra that if a product of several terms is zero, then at least one of those terms must be zero. So, for this equation to hold true, x must be such that (x - 1) = 0, or (x - 3) = 0, or (x - 4) = 0, or (x - 7) = 0. That means x can be 1, 3, 4, or 7. So set A is {1, 3, 4, 7}.Okay, so now I need to find the complement of set A in U, denoted as ∁UA. The complement of a set A in U is all the elements that are in U but not in A. So, I need to list out all the elements in U and then exclude the ones that are in A.Let me write down U again: {0, 1, 2, 3, 4, 5, 6, 7}. And set A is {1, 3, 4, 7}. So, to find the complement, I'll go through each element in U and see if it's not in A.Starting with 0: Is 0 in A? No, because A has 1, 3, 4, 7. So 0 is in the complement.Next, 1: Is 1 in A? Yes, it is. So 1 is not in the complement.Then, 2: Is 2 in A? No, so 2 is in the complement.3: Is 3 in A? Yes, so 3 is not in the complement.4: Is 4 in A? Yes, so 4 is not in the complement.5: Is 5 in A? No, so 5 is in the complement.6: Is 6 in A? No, so 6 is in the complement.7: Is 7 in A? Yes, so 7 is not in the complement.So, putting it all together, the complement of A in U is {0, 2, 5, 6}.Wait, let me double-check to make sure I didn't miss anything. U has 8 elements: 0 through 7. A has 4 elements: 1, 3, 4, 7. The complement should have the remaining 4 elements, which are 0, 2, 5, and 6. Yeah, that seems right.I think I've got it. The complement of set A in the universal set U is {0, 2, 5, 6}.

💡Okay, so I have this problem about an ellipse with its foci at (1, 1) and (1, 5), and it passes through the point (-4, 3). I need to find the equation of the ellipse in the standard form and then determine the sum of a and k, where a is the semi-major axis and k is the y-coordinate of the center.First, I remember that an ellipse is defined as the set of points where the sum of the distances from any point on the ellipse to the two foci is constant. This constant is equal to the length of the major axis, which is 2a, where a is the semi-major axis.So, I need to calculate the sum of the distances from the given point (-4, 3) to each of the foci. Let me write down the coordinates of the foci: (1, 1) and (1, 5). The point on the ellipse is (-4, 3). To find the distance from (-4, 3) to (1, 1), I can use the distance formula: √[(x2 - x1)² + (y2 - y1)²]. Plugging in the values, that would be √[(1 - (-4))² + (1 - 3)²] = √[(5)² + (-2)²] = √[25 + 4] = √29. Similarly, the distance from (-4, 3) to (1, 5) is √[(1 - (-4))² + (5 - 3)²] = √[(5)² + (2)²] = √[25 + 4] = √29. So, both distances are √29. Therefore, the sum of these distances is √29 + √29 = 2√29. This sum is equal to the length of the major axis, so 2a = 2√29, which means a = √29. Wait, hold on, that doesn't seem right. Because in the standard form of an ellipse, a is the semi-major axis, so if the major axis is 2a, then 2a should be the sum of the distances. So, 2a = 2√29, so a = √29. That seems correct.Next, I need to find the center of the ellipse. The center is the midpoint between the two foci. The foci are at (1, 1) and (1, 5). The midpoint formula is [(x1 + x2)/2, (y1 + y2)/2]. Plugging in the values, the x-coordinate is (1 + 1)/2 = 1, and the y-coordinate is (1 + 5)/2 = 3. So, the center of the ellipse is at (1, 3). Therefore, h = 1 and k = 3.Now, I need to determine whether the major axis is vertical or horizontal. Since the foci are aligned vertically (both have the same x-coordinate, 1), the major axis must be vertical. That means the standard form of the ellipse will be [(x - h)² / b²] + [(y - k)² / a²] = 1, where a > b.Wait, actually, I might have that backwards. Let me double-check. In the standard form, if the major axis is vertical, then the denominator under the y-term is larger. So, yes, it should be [(x - h)² / b²] + [(y - k)² / a²] = 1, with a > b.But earlier, I found that a = √29. Is that correct? Wait, let me think again. The sum of the distances is 2a, so 2a = 2√29, so a = √29. That seems correct.But I also need to find b, the semi-minor axis. To find b, I can use the relationship between a, b, and c in an ellipse, where c is the distance from the center to each focus. The formula is a² = b² + c².First, let's find c. The distance from the center to each focus is the distance between (1, 3) and (1, 1), which is |3 - 1| = 2. So, c = 2.Now, plugging into the formula: a² = b² + c². We have a = √29, so a² = 29. c = 2, so c² = 4. Therefore, 29 = b² + 4, which means b² = 25, so b = 5.So, now I have a = √29, b = 5, h = 1, k = 3. Therefore, the equation of the ellipse is [(x - 1)² / 25] + [(y - 3)² / 29] = 1.Wait, but the problem asks for the equation in the form [(x - h)² / a²] + [(y - k)² / b²] = 1. But in this case, since the major axis is vertical, the larger denominator is under the y-term, so actually, a² is under the y-term. So, in the standard form, a is associated with the major axis, which is vertical here.But in the problem statement, the form is given as [(x - h)² / a²] + [(y - k)² / b²] = 1. Hmm, that might be a bit confusing because usually, a is the semi-major axis, but in this case, since the major axis is vertical, the denominator under y is a². So, perhaps in the problem's given form, a is actually the semi-minor axis, and b is the semi-major axis.Wait, that might be the case. Let me check the problem statement again. It says: "its equation can be written in the form [(x - h)² / a²] + [(y - k)² / b²] = 1 where a, b, h, k are constants, and a and b are positive." It doesn't specify whether a or b is the semi-major or semi-minor axis. So, depending on the orientation, a could be the semi-major or semi-minor.But in the standard form, if the major axis is vertical, the larger denominator is under the y-term. So, in this case, since the major axis is vertical, the denominator under y is a², meaning a is the semi-major axis. But in the problem's given form, the denominator under x is a², which would imply that a is the semi-minor axis if the major axis is vertical.Wait, this is confusing. Let me clarify. In the standard form, if the major axis is vertical, the equation is [(x - h)² / b²] + [(y - k)² / a²] = 1, where a > b. So, in this case, a is the semi-major axis, and b is the semi-minor axis.But in the problem, the form is given as [(x - h)² / a²] + [(y - k)² / b²] = 1. So, if the major axis is vertical, then the larger denominator is under y, so b² would be larger than a². Therefore, in this case, b is the semi-major axis, and a is the semi-minor axis.Wait, that makes sense. So, in the problem's given form, if the major axis is vertical, then b > a. So, in our case, since the major axis is vertical, b = √29, and a = 5.Therefore, the equation is [(x - 1)² / 25] + [(y - 3)² / 29] = 1.So, in this equation, a² = 25, so a = 5, and k = 3. Therefore, a + k = 5 + 3 = 8.Wait, but earlier I thought a was √29, but that was under the assumption that a was the semi-major axis. But in the problem's given form, since the major axis is vertical, the semi-major axis is b, and a is the semi-minor axis. So, a = 5, which is the semi-minor axis, and b = √29, which is the semi-major axis.Therefore, the answer is a + k = 5 + 3 = 8.I think that makes sense now. So, the key was to recognize that in the problem's given form, the denominator under x is a², so if the major axis is vertical, then a is the semi-minor axis, and b is the semi-major axis. Therefore, a = 5, and k = 3, so a + k = 8.

💡First, I'll define the current ages of John and Tom as ( J ) and ( T ) respectively.From the first statement, "John was thrice as old as Tom 6 years ago," I can write the equation:[ J - 6 = 3(T - 6) ]From the second statement, "John will be 2 times as old as Tom in 4 years," I can write the equation:[ J + 4 = 2(T + 4) ]Now, I'll simplify both equations to make them easier to solve.Starting with the first equation:[ J - 6 = 3T - 18 ][ J = 3T - 12 ]And the second equation:[ J + 4 = 2T + 8 ][ J = 2T + 4 ]Since both expressions equal ( J ), I can set them equal to each other:[ 3T - 12 = 2T + 4 ]Solving for ( T ):[ 3T - 2T = 4 + 12 ][ T = 16 ]Therefore, Tom is currently 16 years old.

💡First, I need to calculate the expression ((-2b)^3). To do this, I'll apply the rule of exponents which states that when raising a product to a power, each factor in the product is raised to that power.So, ((-2b)^3) can be broken down into ((-2)^3 cdot b^3).Next, I'll calculate ((-2)^3). Multiplying (-2) by itself three times:[-2 times -2 = 4][4 times -2 = -8]So, ((-2)^3 = -8).Then, I'll calculate (b^3), which is simply (b times b times b = b^3).Finally, I'll combine these results:[-8 cdot b^3 = -8b^3]Therefore, the final answer is (-8b^3).

💡Okay, so I have this problem: I need to find the sum of all positive two-digit integers that are divisible by the sum, difference, and product of their digits. Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, let's understand what the problem is asking. A two-digit number can be represented as 10a + b, where 'a' is the tens digit and 'b' is the units digit. So, for example, in the number 24, 'a' is 2 and 'b' is 4. Now, the problem states that this number, 10a + b, must be divisible by three things: the sum of its digits (a + b), the difference of its digits (a - b), and the product of its digits (a * b). So, for each two-digit number, I need to check if 10a + b is divisible by a + b, a - b, and a * b.Let me write down the conditions mathematically:1. (10a + b) must be divisible by (a + b). So, (10a + b) / (a + b) should be an integer.2. (10a + b) must be divisible by (a - b). So, (10a + b) / (a - b) should be an integer.3. (10a + b) must be divisible by (a * b). So, (10a + b) / (a * b) should be an integer.Alright, so I need to find all two-digit numbers where these three conditions hold true.Let me start by considering the first condition: (10a + b) divisible by (a + b). Let's denote S = a + b. So, 10a + b must be a multiple of S. Let's see if I can express this in another way.10a + b = 9a + (a + b) = 9a + S. So, 10a + b = 9a + S. Therefore, 10a + b is equal to 9a + S, which means that S divides 9a + S. But since S divides itself, it must also divide 9a. So, S divides 9a. That gives me that (a + b) divides 9a.Similarly, moving on to the second condition: (10a + b) divisible by (a - b). Let's denote D = a - b. So, 10a + b must be a multiple of D. Let's see if I can express this in another way.10a + b = 11a - (a - b) = 11a - D. So, 10a + b = 11a - D. Therefore, 10a + b is equal to 11a - D, which means that D divides 11a - D. Since D divides itself, it must also divide 11a. So, D divides 11a. That gives me that (a - b) divides 11a.Now, the third condition is that (10a + b) must be divisible by (a * b). Let's denote P = a * b. So, 10a + b must be a multiple of P. That is, (10a + b) / P must be an integer.So, summarizing the three conditions:1. (a + b) divides 9a.2. (a - b) divides 11a.3. (a * b) divides (10a + b).Now, since a and b are digits, a can range from 1 to 9 (since it's a two-digit number), and b can range from 0 to 9.Let me try to find possible values of a and b that satisfy these conditions.Starting with the first condition: (a + b) divides 9a.So, (a + b) | 9a. That means that 9a is a multiple of (a + b). Let me write that as 9a = k(a + b), where k is some integer.So, 9a = k(a + b). Let's rearrange this equation:9a = ka + kb9a - ka = kba(9 - k) = kbSo, a = (kb)/(9 - k)Since a and b are digits, a must be an integer between 1 and 9, and b must be an integer between 0 and 9.Let me consider possible values of k. Since a and b are positive integers, and 9 - k must be positive (to keep a positive), so 9 - k > 0 => k < 9.Also, since k must be a positive integer (because 9a is positive), k can be 1, 2, 3, 4, 5, 6, 7, or 8.Let me try each value of k and see if I can find integer values of a and b.Starting with k = 1:a = (1*b)/(9 - 1) = b/8Since a must be an integer, b must be a multiple of 8. So, b can be 0 or 8.If b = 0:a = 0/8 = 0, but a cannot be 0 because it's a two-digit number. So, this is invalid.If b = 8:a = 8/8 = 1So, a = 1, b = 8. Let's check if this satisfies all three conditions.Number: 10*1 + 8 = 18Sum of digits: 1 + 8 = 9. 18 is divisible by 9. Good.Difference of digits: 1 - 8 = -7. 18 is divisible by -7? Wait, 18 divided by -7 is not an integer. Hmm, that's a problem.Wait, but divisibility by a negative number is the same as divisibility by its absolute value. So, 18 is divisible by 7? 18 divided by 7 is approximately 2.57, which is not an integer. So, this doesn't satisfy the second condition. Therefore, k = 1 doesn't give a valid solution.Next, k = 2:a = (2b)/(9 - 2) = (2b)/7So, 2b must be divisible by 7. Since b is a digit (0-9), let's see which b satisfies this.2b divisible by 7: b must be such that 2b is a multiple of 7. The multiples of 7 up to 18 (since 2*9=18) are 0, 7, 14.So, 2b = 0 => b = 02b = 7 => b = 3.5, which is not an integer.2b = 14 => b = 7So, possible b values are 0 and 7.If b = 0:a = (2*0)/7 = 0, which is invalid.If b = 7:a = (2*7)/7 = 2So, a = 2, b = 7. Let's check the conditions.Number: 27Sum of digits: 2 + 7 = 9. 27 is divisible by 9. Good.Difference of digits: 2 - 7 = -5. 27 is divisible by 5? 27 divided by 5 is 5.4, which is not an integer. So, this doesn't satisfy the second condition. Therefore, k = 2 also doesn't give a valid solution.Moving on to k = 3:a = (3b)/(9 - 3) = (3b)/6 = b/2So, b must be even for a to be an integer.Possible b values: 0, 2, 4, 6, 8Let's check each:b = 0:a = 0/2 = 0, invalid.b = 2:a = 2/2 = 1Number: 12Sum: 1 + 2 = 3. 12 divisible by 3: yes.Difference: 1 - 2 = -1. 12 divisible by 1: yes.Product: 1*2 = 2. 12 divisible by 2: yes.So, 12 is a valid number.b = 4:a = 4/2 = 2Number: 24Sum: 2 + 4 = 6. 24 divisible by 6: yes.Difference: 2 - 4 = -2. 24 divisible by 2: yes.Product: 2*4 = 8. 24 divisible by 8: yes.So, 24 is valid.b = 6:a = 6/2 = 3Number: 36Sum: 3 + 6 = 9. 36 divisible by 9: yes.Difference: 3 - 6 = -3. 36 divisible by 3: yes.Product: 3*6 = 18. 36 divisible by 18: yes.So, 36 is valid.b = 8:a = 8/2 = 4Number: 48Sum: 4 + 8 = 12. 48 divisible by 12: yes.Difference: 4 - 8 = -4. 48 divisible by 4: yes.Product: 4*8 = 32. 48 divisible by 32? 48/32 = 1.5, which is not an integer. So, this fails the third condition.Therefore, for k = 3, we have valid numbers: 12, 24, 36.Moving on to k = 4:a = (4b)/(9 - 4) = (4b)/5So, 4b must be divisible by 5. Since b is a digit, 4b can be 0, 4, 8, 12, 16, 20, 24, 28, 32, 36.Which of these are divisible by 5? 0, 20.So, 4b = 0 => b = 04b = 20 => b = 5Let's check:b = 0:a = 0/5 = 0, invalid.b = 5:a = (4*5)/5 = 4Number: 45Sum: 4 + 5 = 9. 45 divisible by 9: yes.Difference: 4 - 5 = -1. 45 divisible by 1: yes.Product: 4*5 = 20. 45 divisible by 20? 45/20 = 2.25, which is not an integer. So, fails the third condition.Therefore, k = 4 doesn't give a valid solution.Next, k = 5:a = (5b)/(9 - 5) = (5b)/4So, 5b must be divisible by 4. Since b is a digit, 5b can be 0, 5, 10, 15, 20, 25, 30, 35, 40, 45.Which of these are divisible by 4? 0, 20, 40.So, 5b = 0 => b = 05b = 20 => b = 45b = 40 => b = 8Let's check each:b = 0:a = 0/4 = 0, invalid.b = 4:a = (5*4)/4 = 5Number: 54Sum: 5 + 4 = 9. 54 divisible by 9: yes.Difference: 5 - 4 = 1. 54 divisible by 1: yes.Product: 5*4 = 20. 54 divisible by 20? 54/20 = 2.7, which is not an integer. So, fails the third condition.b = 8:a = (5*8)/4 = 10But a must be a single digit (1-9). So, a = 10 is invalid.Therefore, k = 5 doesn't give a valid solution.Moving on to k = 6:a = (6b)/(9 - 6) = (6b)/3 = 2bSo, a = 2b. Since a is a digit (1-9), b can be 1 to 4 (since 2*5=10 is invalid).Let's check each possible b:b = 1:a = 2*1 = 2Number: 21Sum: 2 + 1 = 3. 21 divisible by 3: yes.Difference: 2 - 1 = 1. 21 divisible by 1: yes.Product: 2*1 = 2. 21 divisible by 2? 21/2 = 10.5, which is not an integer. So, fails the third condition.b = 2:a = 2*2 = 4Number: 42Sum: 4 + 2 = 6. 42 divisible by 6: yes.Difference: 4 - 2 = 2. 42 divisible by 2: yes.Product: 4*2 = 8. 42 divisible by 8? 42/8 = 5.25, which is not an integer. So, fails the third condition.b = 3:a = 2*3 = 6Number: 63Sum: 6 + 3 = 9. 63 divisible by 9: yes.Difference: 6 - 3 = 3. 63 divisible by 3: yes.Product: 6*3 = 18. 63 divisible by 18? 63/18 = 3.5, which is not an integer. So, fails the third condition.b = 4:a = 2*4 = 8Number: 84Sum: 8 + 4 = 12. 84 divisible by 12: yes.Difference: 8 - 4 = 4. 84 divisible by 4: yes.Product: 8*4 = 32. 84 divisible by 32? 84/32 = 2.625, which is not an integer. So, fails the third condition.Therefore, k = 6 doesn't give a valid solution.Next, k = 7:a = (7b)/(9 - 7) = (7b)/2So, 7b must be even, meaning b must be even.Possible b values: 0, 2, 4, 6, 8Let's check each:b = 0:a = 0/2 = 0, invalid.b = 2:a = (7*2)/2 = 7Number: 72Sum: 7 + 2 = 9. 72 divisible by 9: yes.Difference: 7 - 2 = 5. 72 divisible by 5? 72/5 = 14.4, which is not an integer. So, fails the second condition.b = 4:a = (7*4)/2 = 14But a must be a single digit, so a = 14 is invalid.b = 6:a = (7*6)/2 = 21Invalid, as a must be single digit.b = 8:a = (7*8)/2 = 28Invalid, as a must be single digit.Therefore, k = 7 doesn't give a valid solution.Finally, k = 8:a = (8b)/(9 - 8) = 8bSo, a = 8b. Since a is a digit (1-9), b can only be 1 (since 8*2=16 is invalid).b = 1:a = 8*1 = 8Number: 81Sum: 8 + 1 = 9. 81 divisible by 9: yes.Difference: 8 - 1 = 7. 81 divisible by 7? 81/7 ≈ 11.57, which is not an integer. So, fails the second condition.Therefore, k = 8 doesn't give a valid solution.So, from the first condition, the only valid numbers we found are 12, 24, and 36.Now, let's check if these numbers also satisfy the second condition, which is that (10a + b) is divisible by (a - b).For 12:a = 1, b = 2a - b = -112 divisible by -1: yes, since 12 / (-1) = -12, which is an integer.For 24:a = 2, b = 4a - b = -224 divisible by -2: yes, 24 / (-2) = -12, which is an integer.For 36:a = 3, b = 6a - b = -336 divisible by -3: yes, 36 / (-3) = -12, which is an integer.So, all three numbers satisfy the second condition.Now, let's check the third condition: (10a + b) must be divisible by (a * b).For 12:a = 1, b = 2a * b = 212 divisible by 2: yes.For 24:a = 2, b = 4a * b = 824 divisible by 8: yes.For 36:a = 3, b = 6a * b = 1836 divisible by 18: yes.So, all three numbers satisfy the third condition as well.Therefore, the numbers that satisfy all three conditions are 12, 24, and 36.Now, the problem asks for the sum of all such numbers. So, let's add them up:12 + 24 + 36 = 72So, the sum is 72.I think that's all. I don't think I missed any numbers because I systematically checked each possible k value and found that only 12, 24, and 36 satisfy all the conditions.

💡First, I need to find the mean, median, and mode of the given numbers.To find the mean, I'll add all the numbers together and then divide by the total number of numbers. The sum of the numbers is 0 + 1 + 1 + 2 + 2 + 2 + 5 + 5 + 5 + 6, which equals 29. Since there are 10 numbers, the mean is 29 divided by 10, which is 2.9.Next, to find the median, I'll arrange the numbers in ascending order and find the middle value. The ordered list is 0, 1, 1, 2, 2, 2, 5, 5, 5, 6. Since there are 10 numbers, the median will be the average of the 5th and 6th numbers. Both the 5th and 6th numbers are 2, so the median is 2.Finally, to find the mode, I'll identify the number that appears most frequently. In the list, the number 5 appears three times, which is more than any other number. Therefore, the mode is 5.Now, I'll add the mean, median, and mode together: 2.9 (mean) + 2 (median) + 5 (mode) equals 9.9.