answer：Okay, so I have this problem where there's a bin with 10 kg of peanuts. Then, they remove 2 kg of peanuts and add 2 kg of raisins, mix it thoroughly. Then, they do it again: remove 2 kg of the mixture and add 2 kg of raisins. I need to find the ratio of the mass of peanuts to raisins in the final mixture. The options are given from A to E, with E being 16:9. Hmm, okay.Let me try to break this down step by step. Maybe I can model each step and see how the quantities change.First, the bin starts with 10 kg of peanuts. So, initially, there are 10 kg of peanuts and 0 kg of raisins.Then, the first operation: remove 2 kg of peanuts and add 2 kg of raisins. So, after this step, the peanuts would be 10 - 2 = 8 kg, and raisins would be 0 + 2 = 2 kg. So, the total mass remains 10 kg.Now, the mixture is 8 kg peanuts and 2 kg raisins. Then, they remove 2 kg of this mixture. Since it's thoroughly mixed, the ratio of peanuts to raisins in the mixture is 8:2, which simplifies to 4:1. So, in the 2 kg removed, how much is peanuts and how much is raisins?To find that, I can think of it as removing 2 kg of the mixture which has 4 parts peanuts and 1 part raisins. So, total parts = 5. Therefore, peanuts removed would be (4/5)*2 kg, and raisins removed would be (1/5)*2 kg.Calculating that: peanuts removed = (4/5)*2 = 8/5 = 1.6 kg, and raisins removed = (1/5)*2 = 0.4 kg.So, after removing 2 kg of the mixture, the remaining peanuts would be 8 - 1.6 = 6.4 kg, and remaining raisins would be 2 - 0.4 = 1.6 kg.Then, they add 2 kg of raisins again. So, the raisins become 1.6 + 2 = 3.6 kg, while peanuts remain at 6.4 kg.So, now, the final mixture has 6.4 kg peanuts and 3.6 kg raisins. The total mass is still 10 kg, which makes sense.Now, the question is asking for the ratio of peanuts to raisins. So, that would be 6.4 : 3.6. Let me simplify that.First, both numbers can be multiplied by 10 to eliminate decimals: 64 : 36.Now, let's see if we can divide both by a common factor. 64 and 36 are both divisible by 4: 64 ÷ 4 = 16, 36 ÷ 4 = 9. So, the ratio simplifies to 16:9.Looking at the options, E is 16:9. So, that should be the answer.Wait, let me double-check my calculations to make sure I didn't make a mistake.Starting with 10 kg peanuts.After first removal and addition: 8 kg peanuts, 2 kg raisins.Then, removing 2 kg of mixture. The ratio is 4:1, so 1.6 kg peanuts and 0.4 kg raisins removed.So, remaining peanuts: 8 - 1.6 = 6.4 kg.Remaining raisins: 2 - 0.4 = 1.6 kg.Then, adding 2 kg raisins: 1.6 + 2 = 3.6 kg.So, final ratio: 6.4 : 3.6.Convert to whole numbers: 64 : 36, which is 16:9.Yes, that seems correct. So, the answer is E.Alternatively, maybe I can think of this in terms of concentration or something else.Another approach: Each time we remove 2 kg of the mixture, which is 20% of the total 10 kg. So, each time, we're removing 20% of the current peanuts and 20% of the current raisins.First step: 10 kg peanuts, 0 kg raisins.After removing 2 kg peanuts and adding 2 kg raisins: 8 kg peanuts, 2 kg raisins.Second step: Remove 20% of the mixture. So, 20% of 8 kg peanuts is 1.6 kg, and 20% of 2 kg raisins is 0.4 kg. So, removing 1.6 kg peanuts and 0.4 kg raisins, then adding 2 kg raisins.So, remaining peanuts: 8 - 1.6 = 6.4 kg.Remaining raisins: 2 - 0.4 + 2 = 3.6 kg.Same result as before. So, the ratio is 6.4:3.6, which is 16:9.Therefore, the answer is E.Wait, just to make sure, maybe I can represent this with equations.Let me denote P as the mass of peanuts and R as the mass of raisins.Initially, P = 10 kg, R = 0 kg.After first operation:P = 10 - 2 = 8 kgR = 0 + 2 = 2 kgThen, when we remove 2 kg of the mixture, the fraction of peanuts in the mixture is P / (P + R) = 8 / 10 = 0.8, and the fraction of raisins is 2 / 10 = 0.2.So, the amount of peanuts removed is 0.8 * 2 kg = 1.6 kg.Similarly, raisins removed is 0.2 * 2 kg = 0.4 kg.So, after removal:P = 8 - 1.6 = 6.4 kgR = 2 - 0.4 = 1.6 kgThen, adding 2 kg raisins:R = 1.6 + 2 = 3.6 kgSo, final P = 6.4 kg, R = 3.6 kg.Ratio P:R = 6.4:3.6 = 16:9.Yep, same result. So, I think that's solid.Alternatively, maybe using the concept of successive dilution or something like that.But in this case, since we are both removing and adding, it's a bit different.Wait, another way: think of it as each time, the concentration of peanuts is being diluted by removing some mixture and adding more raisins.But in this problem, we have two operations, each time removing 2 kg and adding 2 kg raisins.Alternatively, maybe using the formula for concentration after multiple replacements.But since the first operation is not a replacement but a direct removal and addition, maybe it's better to model each step.But as I did above, step by step, seems the most straightforward.So, to recap:Start: 10 kg peanuts, 0 kg raisins.After first step: 8 kg peanuts, 2 kg raisins.After second step: 6.4 kg peanuts, 3.6 kg raisins.Ratio: 6.4:3.6 = 16:9.So, the answer is E.I think that's thorough enough. I considered multiple approaches, double-checked the calculations, and all lead to the same conclusion. So, I feel confident that E is the correct answer.**Final Answer**boxed{E}

answer：Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that the sum of the first n terms is S_n, and specifically, S₁₂ = 21. Then it's asking for the sum of a₂ + a₅ + a₈ + a₁₁. Hmm, okay.First, let me recall what an arithmetic sequence is. It's a sequence where each term after the first is obtained by adding a constant difference. So, if the first term is a₁ and the common difference is d, then the nth term a_n is given by a₁ + (n-1)d. Got that.Now, the sum of the first n terms, S_n, of an arithmetic sequence is given by the formula S_n = n/2 [2a₁ + (n-1)d]. Alternatively, it can also be written as S_n = n(a₁ + a_n)/2. Both formulas are equivalent because a_n = a₁ + (n-1)d. So, that's good to remember.Given that S₁₂ = 21, I can plug that into the formula. Let me write that down:S₁₂ = 12/2 [2a₁ + (12 - 1)d] = 21.Simplifying that, 12 divided by 2 is 6, so:6[2a₁ + 11d] = 21.Divide both sides by 6:2a₁ + 11d = 21/6.Wait, 21 divided by 6 is 3.5, which is 7/2. So:2a₁ + 11d = 7/2.Hmm, okay. So that's one equation involving a₁ and d. But I need to find a₂ + a₅ + a₈ + a₁₁. Let me write expressions for each of these terms.a₂ is the second term, so that's a₁ + d.a₅ is the fifth term, which is a₁ + 4d.a₈ is the eighth term, so a₁ + 7d.a₁₁ is the eleventh term, so a₁ + 10d.So, adding them up:a₂ + a₅ + a₈ + a₁₁ = (a₁ + d) + (a₁ + 4d) + (a₁ + 7d) + (a₁ + 10d).Let me combine like terms. There are four a₁ terms and then the d terms:4a₁ + (1 + 4 + 7 + 10)d.Calculating the coefficients:1 + 4 is 5, 5 + 7 is 12, 12 + 10 is 22. So, 22d.So, the sum is 4a₁ + 22d.Hmm, okay. So, I have 4a₁ + 22d. I need to find the value of this expression. From earlier, I have 2a₁ + 11d = 7/2. Wait, 4a₁ + 22d is just 2*(2a₁ + 11d). Let me check:2*(2a₁ + 11d) = 4a₁ + 22d. Yes, exactly.So, since 2a₁ + 11d = 7/2, then 4a₁ + 22d is 2*(7/2) = 7.Oh! So, the sum a₂ + a₅ + a₈ + a₁₁ is 7.Wait, that seems straightforward. Let me just verify to make sure I didn't skip any steps or make a mistake.Starting again, S₁₂ = 21. So, 12/2 [2a₁ + 11d] = 21. Simplify: 6*(2a₁ + 11d) = 21. So, 2a₁ + 11d = 21/6 = 3.5 = 7/2. Then, a₂ + a₅ + a₈ + a₁₁ is 4a₁ + 22d, which is 2*(2a₁ + 11d) = 2*(7/2) = 7. Yep, that seems correct.Alternatively, another way to think about it is that the terms a₂, a₅, a₈, a₁₁ are equally spaced in the sequence. Specifically, each term is 3 positions apart. So, starting from a₂, then a₅ is 3 terms later, a₈ is another 3 terms, and a₁₁ is another 3 terms. So, in an arithmetic sequence, the average of equally spaced terms is equal to the average of the first and last term in that subset.So, the average of a₂, a₅, a₈, a₁₁ would be (a₂ + a₁₁)/2. Since there are four terms, the sum would be 4*(a₂ + a₁₁)/2 = 2*(a₂ + a₁₁).Wait, let me compute a₂ + a₁₁. a₂ is a₁ + d, a₁₁ is a₁ + 10d. So, a₂ + a₁₁ = 2a₁ + 11d. Which is exactly the same as before. So, 2*(2a₁ + 11d) = 4a₁ + 22d, which is 7.So, that's another way to see it. So, whether I compute each term individually and sum them up or use the property of equally spaced terms in an arithmetic sequence, I get the same result.Therefore, I can be confident that the sum a₂ + a₅ + a₈ + a₁₁ is 7.**Final Answer**The sum (a_{2} + a_{5} + a_{8} + a_{11}) is boxed{7}.

answer：First, I need to determine the total number of pies Grandma baked. She made 21 batches, each containing ( N ) pies, so the total number of pies is ( 21N ).Next, since Grandma distributed all the pies onto trays with 70 pies each, the total number of pies must be divisible by 70. This means that ( 21N ) must be a multiple of 70.To find the smallest possible value of ( N ) that satisfies this condition, I'll set up the equation:[21N = 70k]where ( k ) is an integer representing the number of trays.Simplifying this equation, I divide both sides by 7:[3N = 10k]This implies that ( N ) must be a multiple of 10. Let's express ( N ) as:[N = 10m]where ( m ) is an integer.Substituting back into the equation:[3(10m) = 10k implies 30m = 10k implies 3m = k]This means that ( k ) must be a multiple of 3. The smallest positive integer value for ( m ) is 1, which makes ( k = 3 ).Therefore, the smallest possible value of ( N ) is:[N = 10 times 1 = 10]However, the problem states that ( N > 70 ). Since 10 is less than 70, I need to find the next multiple of 10 that satisfies ( N > 70 ).The smallest multiple of 10 greater than 70 is 70 itself, but since ( N ) must be greater than 70, the next multiple is 80.Thus, the smallest possible value of ( N ) that meets all the conditions is 80.

answer：Okay, so I have this problem here about an arithmetic sequence. It says that the sum of the first n terms is S_n, and specifically, S_9 is 27. I need to find a_4 + a_6. Hmm, arithmetic sequences, okay, let me recall what I know about them.First, an arithmetic sequence is a sequence where each term increases by a constant difference. So, if the first term is a_1 and the common difference is d, then the nth term a_n can be written as a_1 + (n-1)d. Got that part down.Now, the sum of the first n terms, S_n, of an arithmetic sequence has a formula too. I think it's S_n = n/2 * (a_1 + a_n). Alternatively, it can also be written as S_n = n/2 * [2a_1 + (n-1)d]. Yeah, that sounds right. So, both formulas are equivalent because a_n is a_1 + (n-1)d.Given that S_9 = 27, let me write that down using the sum formula. So, S_9 = 9/2 * [2a_1 + (9-1)d] = 27. Simplifying that, 9/2 * [2a_1 + 8d] = 27. Let me compute that step by step.First, 9/2 multiplied by something equals 27. So, if I let that something be [2a_1 + 8d], then 9/2 * [2a_1 + 8d] = 27. To solve for [2a_1 + 8d], I can multiply both sides by 2/9. So, [2a_1 + 8d] = 27 * (2/9). Let me calculate that: 27 divided by 9 is 3, times 2 is 6. So, 2a_1 + 8d = 6.Hmm, okay, so 2a_1 + 8d = 6. Maybe I can simplify that equation. Let's see, I can factor out a 2 from the left side: 2(a_1 + 4d) = 6. Then, dividing both sides by 2 gives a_1 + 4d = 3.Wait a second, a_1 + 4d is actually the fifth term of the arithmetic sequence, right? Because a_n = a_1 + (n-1)d, so a_5 = a_1 + 4d. So, a_5 = 3. Interesting, so the fifth term is 3.But the question is asking for a_4 + a_6. Let me write expressions for a_4 and a_6. a_4 = a_1 + 3da_6 = a_1 + 5dSo, adding them together: a_4 + a_6 = (a_1 + 3d) + (a_1 + 5d) = 2a_1 + 8d.Wait, that's the same expression I had earlier! 2a_1 + 8d. And from earlier, I found that 2a_1 + 8d = 6. So, that means a_4 + a_6 = 6.Wait, so is that the answer? 6? Hmm, let me double-check.Alternatively, since a_4 + a_6 is equal to 2a_5 because in an arithmetic sequence, the terms equidistant from the center add up to twice the middle term. Since 4 and 6 are equidistant from 5, their sum is twice a_5. And since a_5 is 3, then 2*3=6. So, that confirms it.So, yeah, either way I look at it, a_4 + a_6 is 6.**Final Answer**The value of (a_4 + a_6) is boxed{6}.

answer：Alright, so I came across this interesting problem about "good" natural numbers. Let me try to wrap my head around it step by step. The problem says that a natural number greater than 1 is called "good" if it's exactly equal to the product of its distinct proper divisors, excluding 1 and the number itself. Then, I need to find the sum of the first ten such "good" numbers.First off, let me make sure I understand what a proper divisor is. A proper divisor of a number is a divisor other than the number itself. But in this case, we're also excluding 1. So, for a number n, we're looking at divisors d where 1 < d < n.Now, the key part is that the number n should be equal to the product of its distinct proper divisors (excluding 1 and itself). Hmm, okay. So, for example, if n is a prime number, it doesn't have any proper divisors other than 1 and itself, so it wouldn't have any divisors to consider here. Therefore, prime numbers can't be "good" numbers because they don't have any proper divisors besides 1 and themselves.So, the first set of numbers we can consider are composite numbers. Let me think about the smallest composite numbers and see if they fit the criteria.Starting with 4. The proper divisors of 4, excluding 1 and 4, are just 2. So, the product of its proper divisors is 2. But 2 is not equal to 4, so 4 isn't a "good" number.Next, 6. The proper divisors of 6, excluding 1 and 6, are 2 and 3. The product of these is 2*3=6. Hey, that's equal to the number itself! So, 6 is a "good" number. That's our first one.Moving on to 8. The proper divisors of 8, excluding 1 and 8, are 2 and 4. The product is 2*4=8. That's equal to 8, so 8 is also a "good" number. That's our second one.Next, 9. The proper divisors of 9, excluding 1 and 9, are just 3. The product is 3, which isn't equal to 9, so 9 isn't good.How about 10? The proper divisors are 2 and 5. Their product is 2*5=10, which equals the number itself. So, 10 is good. That's our third.12 comes next. The proper divisors of 12, excluding 1 and 12, are 2, 3, 4, and 6. The product is 2*3*4*6. Let me calculate that: 2*3=6, 6*4=24, 24*6=144. 144 is way bigger than 12, so 12 isn't good.Hmm, interesting. So, 12 doesn't work. Let's try 14. Proper divisors are 2 and 7. Their product is 2*7=14, which equals the number. So, 14 is good. That's our fourth.15: Proper divisors are 3 and 5. 3*5=15, so 15 is good. Fifth one.16: Proper divisors are 2, 4, and 8. The product is 2*4*8=64, which is not equal to 16. So, 16 isn't good.18: Proper divisors are 2, 3, 6, 9. Their product is 2*3*6*9. Let's compute: 2*3=6, 6*6=36, 36*9=324. Not equal to 18. So, 18 isn't good.20: Proper divisors are 2, 4, 5, 10. Their product is 2*4*5*10. 2*4=8, 8*5=40, 40*10=400. Not equal to 20. Not good.21: Proper divisors are 3 and 7. 3*7=21. So, 21 is good. Sixth one.22: Proper divisors are 2 and 11. 2*11=22. So, 22 is good. Seventh.24: Proper divisors are 2, 3, 4, 6, 8, 12. Product is 2*3*4*6*8*12. That's going to be a huge number. Let me compute step by step: 2*3=6, 6*4=24, 24*6=144, 144*8=1152, 1152*12=13824. Definitely not equal to 24. So, 24 isn't good.25: Proper divisors are just 5. 5 is not equal to 25, so 25 isn't good.26: Proper divisors are 2 and 13. 2*13=26. So, 26 is good. Eighth.27: Proper divisors are 3 and 9. 3*9=27. So, 27 is good. Ninth.28: Proper divisors are 2, 4, 7, 14. Their product is 2*4*7*14. Let's compute: 2*4=8, 8*7=56, 56*14=784. Not equal to 28. So, 28 isn't good.30: Proper divisors are 2, 3, 5, 6, 10, 15. Product is 2*3*5*6*10*15. That's going to be massive. Let me compute: 2*3=6, 6*5=30, 30*6=180, 180*10=1800, 1800*15=27000. Not equal to 30. So, 30 isn't good.32: Proper divisors are 2, 4, 8, 16. Product is 2*4*8*16. 2*4=8, 8*8=64, 64*16=1024. Not equal to 32. Not good.33: Proper divisors are 3 and 11. 3*11=33. So, 33 is good. Tenth one.Wait, hold on. So, let me list out the numbers I found so far:6, 8, 10, 14, 15, 21, 22, 26, 27, 33.That's ten numbers. Let me verify each of them to make sure they're correct.6: Divisors are 2,3. 2*3=6. Correct.8: Divisors are 2,4. 2*4=8. Correct.10: Divisors are 2,5. 2*5=10. Correct.14: Divisors are 2,7. 2*7=14. Correct.15: Divisors are 3,5. 3*5=15. Correct.21: Divisors are 3,7. 3*7=21. Correct.22: Divisors are 2,11. 2*11=22. Correct.26: Divisors are 2,13. 2*13=26. Correct.27: Divisors are 3,9. 3*9=27. Correct.33: Divisors are 3,11. 3*11=33. Correct.Okay, so all of these check out. So, the first ten "good" numbers are 6, 8, 10, 14, 15, 21, 22, 26, 27, and 33.Now, to find the sum, I need to add these numbers together.Let me compute step by step:Start with 6.6 + 8 = 14.14 + 10 = 24.24 + 14 = 38.38 + 15 = 53.53 + 21 = 74.74 + 22 = 96.96 + 26 = 122.122 + 27 = 149.149 + 33 = 182.So, the sum of the first ten "good" numbers is 182.Wait, let me double-check the addition to make sure I didn't make a mistake.6 + 8 = 14.14 + 10 = 24.24 + 14 = 38.38 + 15 = 53.53 + 21 = 74.74 + 22 = 96.96 + 26 = 122.122 + 27 = 149.149 + 33 = 182.Yes, that seems correct.But just to be thorough, let me add them in a different order to cross-verify.Let me pair them:6 and 33: 6 + 33 = 39.8 and 27: 8 + 27 = 35.10 and 26: 10 + 26 = 36.14 and 22: 14 + 22 = 36.15 and 21: 15 + 21 = 36.Now, adding these pairs: 39 + 35 = 74; 74 + 36 = 110; 110 + 36 = 146; 146 + 36 = 182.Same result. So, that seems consistent.Alternatively, adding all together:6 + 8 + 10 + 14 + 15 + 21 + 22 + 26 + 27 + 33.Let me group them:(6 + 33) + (8 + 27) + (10 + 26) + (14 + 22) + (15 + 21).Which is 39 + 35 + 36 + 36 + 36.Adding these: 39 + 35 = 74; 74 + 36 = 110; 110 + 36 = 146; 146 + 36 = 182.Same answer again.So, I feel confident that the sum is 182.But just to make sure I didn't miss any "good" numbers before 33, let me check the numbers between 27 and 33.28: Not good, as we saw earlier.29: Prime, so not good.30: Not good.31: Prime, so not good.32: Not good.33: Good.So, yes, 33 is the tenth "good" number.Therefore, the sum is 182.**Final Answer**The sum of the first ten "good" natural numbers is boxed{182}.

answer：Okay, so I have this problem here: "The distance from a point M on the parabola y² = 4x to its focus is 5. Find the x-coordinate of point M." Hmm, let me try to figure this out step by step.First, I remember that the standard form of a parabola that opens to the right is y² = 4ax, where 'a' is the distance from the vertex to the focus. Comparing this with the given equation y² = 4x, I can see that 4a = 4, so a = 1. That means the focus of this parabola is at (a, 0), which is (1, 0). Got that down.Now, the problem says that the distance from point M on the parabola to the focus is 5. So, point M is somewhere on the parabola y² = 4x, and the distance between M and the focus (1, 0) is 5 units. I need to find the x-coordinate of M.Let me denote the coordinates of point M as (x, y). Since M lies on the parabola y² = 4x, it must satisfy this equation. So, y² = 4x. That's one equation.The distance between M(x, y) and the focus (1, 0) is given by the distance formula: √[(x - 1)² + (y - 0)²] = 5. So, squaring both sides to eliminate the square root, I get (x - 1)² + y² = 25. That's my second equation.Now, I have two equations:1. y² = 4x2. (x - 1)² + y² = 25I can substitute equation 1 into equation 2 to eliminate y². Let's do that.Substituting y² from equation 1 into equation 2:(x - 1)² + 4x = 25Let me expand (x - 1)²:(x² - 2x + 1) + 4x = 25Now, combine like terms:x² - 2x + 1 + 4x = 25Simplify the x terms:x² + 2x + 1 = 25Hmm, so x² + 2x + 1 = 25. That looks like a quadratic equation. Let me write it as:x² + 2x + 1 - 25 = 0Simplify:x² + 2x - 24 = 0Okay, so I have a quadratic equation: x² + 2x - 24 = 0. I can solve this using the quadratic formula. The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a), where a = 1, b = 2, c = -24.Plugging in the values:x = [-2 ± √( (2)² - 4*1*(-24) )] / (2*1)x = [-2 ± √(4 + 96)] / 2x = [-2 ± √100] / 2x = [-2 ± 10] / 2So, that gives two solutions:1. x = (-2 + 10)/2 = 8/2 = 42. x = (-2 - 10)/2 = (-12)/2 = -6Wait a minute, x = 4 and x = -6. But hold on, the parabola y² = 4x is defined for x ≥ 0 because y² is always non-negative, and 4x must also be non-negative. So, x cannot be negative. Therefore, x = -6 is not a valid solution.So, the only valid solution is x = 4. Therefore, the x-coordinate of point M is 4.But just to make sure I didn't make any mistakes, let me verify this.If x = 4, then from the equation y² = 4x, y² = 16, so y = ±4. So, point M can be (4, 4) or (4, -4). Let me check the distance from (4, 4) to the focus (1, 0):Distance = √[(4 - 1)² + (4 - 0)²] = √[3² + 4²] = √[9 + 16] = √25 = 5. That's correct.Similarly, for (4, -4):Distance = √[(4 - 1)² + (-4 - 0)²] = √[3² + (-4)²] = √[9 + 16] = √25 = 5. Also correct.So, both points are valid, but the x-coordinate is 4 in both cases. Therefore, the answer is 4.I think that's solid. I don't see any mistakes in my reasoning. I considered both solutions from the quadratic, checked their validity based on the parabola's domain, and verified the distance. Everything checks out.**Final Answer**The x-coordinate of point ( M ) is boxed{4}.