💡Okay, so I have this problem: For how many positive integers ( n ) does ( 1 + 2 + cdots + n ) evenly divide ( 8n )? Hmm, let me try to figure this out step by step.First, I remember that the sum of the first ( n ) positive integers is given by the formula ( S_n = frac{n(n+1)}{2} ). So, the problem is asking for how many positive integers ( n ) does ( S_n ) divide ( 8n ) evenly. That means ( frac{8n}{S_n} ) should be an integer.Let me write that down:[frac{8n}{frac{n(n+1)}{2}} = frac{16}{n+1}]Wait, how did I get that? Let me check. If I have ( 8n ) divided by ( frac{n(n+1)}{2} ), that's the same as multiplying ( 8n ) by ( frac{2}{n(n+1)} ), right? So:[8n times frac{2}{n(n+1)} = frac{16n}{n(n+1)} = frac{16}{n+1}]Yes, that makes sense. So, ( frac{16}{n+1} ) needs to be an integer. That means ( n+1 ) must be a divisor of 16.Alright, let's list the positive divisors of 16. They are 1, 2, 4, 8, and 16. So, ( n+1 ) can be 1, 2, 4, 8, or 16. Therefore, ( n ) can be 0, 1, 3, 7, or 15.But the problem specifies positive integers ( n ), so we exclude 0. That leaves us with ( n = 1, 3, 7, 15 ).Let me verify each of these to make sure they work.1. For ( n = 1 ): - Sum ( S_1 = 1 ) - ( 8n = 8 times 1 = 8 ) - ( 8 div 1 = 8 ), which is an integer. Good.2. For ( n = 3 ): - Sum ( S_3 = 1 + 2 + 3 = 6 ) - ( 8n = 8 times 3 = 24 ) - ( 24 div 6 = 4 ), which is an integer. Good.3. For ( n = 7 ): - Sum ( S_7 = 1 + 2 + cdots + 7 = 28 ) - ( 8n = 8 times 7 = 56 ) - ( 56 div 28 = 2 ), which is an integer. Good.4. For ( n = 15 ): - Sum ( S_{15} = 1 + 2 + cdots + 15 = 120 ) - ( 8n = 8 times 15 = 120 ) - ( 120 div 120 = 1 ), which is an integer. Good.So, all four values of ( n ) work. Therefore, there are 4 positive integers ( n ) that satisfy the condition.Wait, just to make sure I didn't miss anything, let me think again. The key was recognizing that ( n+1 ) must divide 16. The divisors of 16 are 1, 2, 4, 8, 16, so ( n ) would be 0, 1, 3, 7, 15. Since ( n ) must be positive, we exclude 0, so 4 values. Yep, that seems right.I don't think I missed any other divisors because 16 is a power of 2, so its divisors are straightforward. Also, checking each ( n ) confirmed that they all work. So, I feel confident that the answer is 4.**Final Answer**The number of positive integers ( n ) is boxed{4}.

💡First, let's denote the complex number ( z ) as ( z = a + bi ), where ( a ) and ( b ) are real numbers.Given the equation ( iz = 1 + 2i ), substitute ( z ) with ( a + bi ):[i(a + bi) = 1 + 2i]Simplify the left side:[ai + b i^2 = ai - b]Since ( i^2 = -1 ).Now, equate the real and imaginary parts:[- b = 1 quad text{and} quad a = 2]From the real part equation:[- b = 1 implies b = -1]So, the complex number ( z ) is:[z = 2 - i]The conjugate of ( z ), denoted as ( overline{z} ), is:[overline{z} = 2 + i]The imaginary part of ( overline{z} ) is ( 1 ).

💡Okay, so I have this problem here: I need to find the greatest possible value of ( x + y ) given that ( x^2 + y^2 = 98 ) and ( xy = 36 ). Hmm, let me think about how to approach this.First off, I remember that there's a formula that relates ( (x + y)^2 ) to ( x^2 + y^2 ) and ( xy ). It goes something like this: ( (x + y)^2 = x^2 + y^2 + 2xy ). Yeah, that sounds right. So maybe I can use that to find ( x + y ).Let me write that down:[(x + y)^2 = x^2 + y^2 + 2xy]Alright, I know ( x^2 + y^2 ) is 98, and ( xy ) is 36. So substituting those values in:[(x + y)^2 = 98 + 2 times 36]Calculating the right side:[2 times 36 = 72][98 + 72 = 170]So,[(x + y)^2 = 170]To find ( x + y ), I need to take the square root of both sides. That gives me:[x + y = sqrt{170} quad text{or} quad x + y = -sqrt{170}]But the problem asks for the greatest possible value of ( x + y ). Between ( sqrt{170} ) and ( -sqrt{170} ), clearly ( sqrt{170} ) is the larger one. So, that should be the answer.Wait, let me double-check if I did everything correctly. I used the identity correctly, substituted the given values, and simplified. Yeah, that seems right. So, the greatest possible value of ( x + y ) is ( sqrt{170} ).I wonder if there's another way to approach this problem, maybe by solving for ( x ) and ( y ) individually. Let me try that just to confirm.Given ( x^2 + y^2 = 98 ) and ( xy = 36 ), maybe I can express one variable in terms of the other and substitute. Let's say I solve for ( y ) from the second equation:[y = frac{36}{x}]Now, substitute this into the first equation:[x^2 + left( frac{36}{x} right)^2 = 98]Simplify that:[x^2 + frac{1296}{x^2} = 98]Hmm, this looks a bit complicated. Maybe I can multiply both sides by ( x^2 ) to eliminate the denominator:[x^4 + 1296 = 98x^2]Rearranging terms:[x^4 - 98x^2 + 1296 = 0]This is a quartic equation, but it's quadratic in terms of ( x^2 ). Let me set ( z = x^2 ), so the equation becomes:[z^2 - 98z + 1296 = 0]Now, I can solve this quadratic equation for ( z ). Using the quadratic formula:[z = frac{98 pm sqrt{98^2 - 4 times 1 times 1296}}{2}]Calculating the discriminant:[98^2 = 9604][4 times 1 times 1296 = 5184][sqrt{9604 - 5184} = sqrt{4420}]Wait, ( 4420 ) doesn't look like a perfect square. Let me check my calculations again. Maybe I made a mistake earlier.Going back to the substitution step:[x^2 + left( frac{36}{x} right)^2 = 98][x^2 + frac{1296}{x^2} = 98][x^4 + 1296 = 98x^2][x^4 - 98x^2 + 1296 = 0]Yes, that seems correct. So, ( z^2 - 98z + 1296 = 0 ). Let me compute the discriminant again:[98^2 = 9604][4ac = 4 times 1 times 1296 = 5184][sqrt{9604 - 5184} = sqrt{4420}]Hmm, ( 4420 ) is not a perfect square. Let me see if I can factor it or simplify it somehow. Breaking down 4420:[4420 = 4 times 1105][1105 = 5 times 221][221 = 13 times 17]So, ( 4420 = 4 times 5 times 13 times 17 ). There's no square factor beyond 4, so ( sqrt{4420} = 2sqrt{1105} ). That doesn't seem helpful.Maybe I made a mistake in the substitution approach. Let me try another way. Alternatively, I could use the identity ( (x - y)^2 = x^2 + y^2 - 2xy ) to find ( x - y ), but I'm not sure if that helps directly.Wait, but I already found ( x + y = sqrt{170} ) using the first method, and that seems straightforward. Maybe the substitution method is just more complicated and not necessary here. Perhaps I should stick with the first solution.Let me verify if ( x + y = sqrt{170} ) satisfies both original equations. If ( x + y = sqrt{170} ) and ( xy = 36 ), then the quadratic equation with roots ( x ) and ( y ) would be:[t^2 - (x + y)t + xy = 0][t^2 - sqrt{170}t + 36 = 0]Calculating the discriminant:[(sqrt{170})^2 - 4 times 1 times 36 = 170 - 144 = 26]So, the roots are:[t = frac{sqrt{170} pm sqrt{26}}{2}]These are real numbers, so ( x ) and ( y ) exist. Therefore, my solution ( x + y = sqrt{170} ) is valid.I think I'm confident with the first method. It was more straightforward and led me to the correct answer without getting bogged down in complicated equations. The substitution method, while possible, introduced a quartic equation which was more difficult to solve, especially since the discriminant wasn't a perfect square. So, I'll stick with ( sqrt{170} ) as the greatest possible value of ( x + y ).Just to recap, the key steps were:1. Recognize the identity ( (x + y)^2 = x^2 + y^2 + 2xy ).2. Substitute the given values ( x^2 + y^2 = 98 ) and ( xy = 36 ).3. Calculate ( (x + y)^2 = 98 + 72 = 170 ).4. Take the square root to find ( x + y = sqrt{170} ) or ( x + y = -sqrt{170} ).5. Choose the positive value since it's the greatest possible.This approach was efficient and avoided unnecessary complexity. I think I've covered all bases here and can confidently say that ( sqrt{170} ) is the correct answer.**Final Answer**The greatest possible value of ( x + y ) is boxed{sqrt{170}}.

💡Alright, so I've got this problem about life insurance and probability. It says that the probability a policyholder in a certain city will live to be 75 years old is 0.60. There are three parts to the problem:1. The probability that all three policyholders live to be 75 years old.2. The probability that only one out of three policyholders lives to be 75 years old.3. The probability that at least one out of three policyholders lives to be 75 years old.Okay, let's start with the first part. I think this is about finding the probability that all three policyholders live to 75. Since each policyholder's survival is an independent event, I can use the multiplication rule for independent events. So, if the probability for one person is 0.60, then for three people, it should be 0.60 multiplied by itself three times, right? That would be 0.60 * 0.60 * 0.60. Let me calculate that: 0.60 * 0.60 is 0.36, and then 0.36 * 0.60 is 0.216. So, the probability that all three live to 75 is 0.216.Moving on to the second part: the probability that only one out of three policyholders lives to be 75. Hmm, this sounds like a binomial probability problem. In binomial problems, we have a fixed number of independent trials, each with two possible outcomes: success or failure. Here, success is living to 75, which has a probability of 0.60, and failure is not living to 75, with a probability of 1 - 0.60 = 0.40.The formula for the probability of exactly k successes in n trials is:P(k) = C(n, k) * p^k * (1-p)^(n-k)Where C(n, k) is the combination of n things taken k at a time. So, for this problem, n = 3, k = 1, p = 0.60.First, let's calculate C(3, 1). That's the number of ways to choose 1 success out of 3 trials. C(3, 1) is 3.Then, p^k is 0.60^1, which is 0.60.And (1-p)^(n-k) is 0.40^(3-1) = 0.40^2 = 0.16.Now, multiply all these together: 3 * 0.60 * 0.16.Let's do the multiplication step by step. 3 * 0.60 is 1.80, and 1.80 * 0.16 is 0.288.So, the probability that only one out of three policyholders lives to be 75 is 0.288.Now, the third part: the probability that at least one out of three policyholders lives to be 75. "At least one" means one or more. Calculating this directly would involve adding the probabilities of exactly one, exactly two, and exactly three policyholders living to 75. But that seems like a lot of work. Maybe there's a smarter way.I remember that in probability, sometimes it's easier to calculate the complement of an event and then subtract it from 1. The complement of "at least one" is "none." So, if I can find the probability that none of the three policyholders live to 75, then subtract that from 1, I'll have the probability that at least one does.The probability that one policyholder does not live to 75 is 0.40. Since the events are independent, the probability that all three do not live to 75 is 0.40 * 0.40 * 0.40 = 0.064.So, the probability that at least one lives to 75 is 1 - 0.064 = 0.936.Let me double-check these calculations to make sure I didn't make any mistakes.For the first part: 0.60^3 = 0.216. That seems right.For the second part: C(3,1) = 3, 0.60^1 = 0.60, 0.40^2 = 0.16. Multiplying them together: 3 * 0.60 * 0.16 = 0.288. That also looks correct.For the third part: Calculating the complement, 0.40^3 = 0.064, then 1 - 0.064 = 0.936. That makes sense too.I think I've got all the parts right. It's always good to verify, especially when dealing with probabilities, because it's easy to mix up combinations or exponents.Another way to think about the third part is to realize that "at least one" is the opposite of "none." So, by calculating the probability of the opposite event and subtracting it from 1, I can efficiently find the desired probability without having to calculate multiple scenarios.Overall, I feel confident about these answers. Probability can sometimes be tricky, especially with combinatorics involved, but breaking it down step by step helps prevent errors.

💡Okay, so I have this quadratic equation: 3x² + 7x + b = 0. I need to find all positive integer values of b such that the equation has rational roots. Then, I have to sum those b values. Hmm, okay, let's think about how to approach this.First, I remember that for a quadratic equation ax² + bx + c = 0, the roots are rational if and only if the discriminant is a perfect square. The discriminant is given by D = b² - 4ac. So, in this case, the discriminant would be D = 7² - 4*3*b. Let me compute that.So, D = 49 - 12b. For the roots to be rational, D must be a perfect square. Also, since we're dealing with real roots, D must be non-negative. Therefore, 49 - 12b ≥ 0. Let me solve this inequality for b.49 - 12b ≥ 0 => -12b ≥ -49 Dividing both sides by -12 (and remembering to flip the inequality sign because I'm dividing by a negative number): b ≤ 49/12 Calculating 49 divided by 12: 49 ÷ 12 is approximately 4.083. So, b must be less than or equal to approximately 4.083. Since b is a positive integer, the possible values of b are 1, 2, 3, and 4.Now, I need to check each of these values of b to see if the discriminant becomes a perfect square.Starting with b = 1: D = 49 - 12*1 = 49 - 12 = 37. 37 is not a perfect square because 6² = 36 and 7² = 49, so 37 is between them but isn't a square. So, b = 1 doesn't work.Next, b = 2: D = 49 - 12*2 = 49 - 24 = 25. 25 is a perfect square because 5² = 25. So, b = 2 is good.Moving on to b = 3: D = 49 - 12*3 = 49 - 36 = 13. 13 isn't a perfect square either; 3² = 9 and 4² = 16, so 13 is in between. Therefore, b = 3 doesn't work.Lastly, b = 4: D = 49 - 12*4 = 49 - 48 = 1. 1 is a perfect square because 1² = 1. So, b = 4 is also good.So, the values of b that satisfy the condition are 2 and 4. Now, I need to find the sum of these values. Let's add them up: 2 + 4 = 6.Therefore, the sum of all possible positive integer values of b is 6.**Final Answer**The sum of all possible positive integer values of ( b ) is boxed{6}.

💡First, I need to determine the measure of arc ( TQ ) that corresponds to the central angle ( angle TOQ ). Since ( angle TOQ ) is a central angle, it directly measures the arc ( TQ ). Therefore, arc ( TQ ) is ( 45^circ ).Next, I'll calculate the circumference of the circle using the radius ( OT = 15 ) cm. The formula for the circumference is:[text{Circumference} = 2pi times text{radius} = 2pi times 15 = 30pi text{ cm}]Now, I'll find the fraction of the circumference that corresponds to the ( 45^circ ) arc. Since a full circle is ( 360^circ ), the fraction is:[frac{45^circ}{360^circ} = frac{1}{8}]Finally, I'll calculate the length of arc ( TQ ) by multiplying the circumference by this fraction:[text{Length of arc } TQ = frac{1}{8} times 30pi = frac{30pi}{8} = frac{15pi}{4} text{ cm}]