💡Alright, let's try to figure out who stole what in this case. So, we have three characters: the March Hare, the Hatter, and the Dormouse. Each of them made a statement during the trial:- March Hare said, "The Hatter stole the butter."- Hatter said, "The Dormouse stole the eggs."- Dormouse said, "I stole the milk."From the problem, we know two important things:1. The one who stole the butter was telling the truth.2. The one who stole the eggs was lying.Okay, let's break this down step by step.First, let's consider the March Hare's statement: "The Hatter stole the butter." If the March Hare is telling the truth, then the Hatter did steal the butter. But if the March Hare is lying, then the Hatter did not steal the butter.Next, the Hatter's statement: "The Dormouse stole the eggs." If the Hatter is telling the truth, then the Dormouse did steal the eggs. If the Hatter is lying, then the Dormouse did not steal the eggs.Lastly, the Dormouse's statement: "I stole the milk." If the Dormouse is telling the truth, then he did steal the milk. If he's lying, then he didn't steal the milk.Now, we know that the one who stole the butter was telling the truth. So, whoever stole the butter must be telling the truth. Also, the one who stole the eggs was lying, meaning whoever stole the eggs must be lying.Let's consider the possibilities.1. Suppose the March Hare stole the butter. Then, according to the first point, he must be telling the truth. So, his statement "The Hatter stole the butter" would be true. But wait, if the March Hare stole the butter, then the Hatter did not steal the butter. This creates a contradiction because the March Hare cannot both steal the butter and truthfully say that the Hatter stole it. Therefore, the March Hare did not steal the butter.2. Next, suppose the Hatter stole the butter. If the Hatter stole the butter, then he must be telling the truth. So, his statement "The Dormouse stole the eggs" would be true. That would mean the Dormouse did steal the eggs. But from the problem, we know that the one who stole the eggs was lying. If the Dormouse stole the eggs, he must be lying. However, if the Hatter is telling the truth, then the Dormouse's statement "I stole the milk" would be a lie, meaning he didn't steal the milk. But if the Dormouse stole the eggs, then he must have stolen the eggs, and the milk must have been stolen by someone else. But we only have three items: butter, eggs, and milk. If the Hatter stole the butter, the Dormouse stole the eggs, then the March Hare must have stolen the milk. But the March Hare's statement was "The Hatter stole the butter," which would be true if the Hatter did steal the butter. But we already assumed the Hatter stole the butter, so the March Hare is telling the truth. However, the March Hare would then be the one who stole the milk, but he didn't make any statement about stealing milk. This seems a bit confusing. Let's try to map it out.- If Hatter stole butter: Hatter tells truth.- Hatter's statement: Dormouse stole eggs (truth).- Therefore, Dormouse stole eggs.- But the one who stole eggs was lying, so Dormouse is lying.- Dormouse's statement: "I stole the milk" is a lie, so he didn't steal milk.- Therefore, milk must have been stolen by March Hare.- March Hare's statement: "Hatter stole butter" is true, so March Hare is telling the truth.- But March Hare stole milk, and he is telling the truth, which is consistent.Wait, this seems to fit. Let's check:- Hatter stole butter: tells truth.- Hatter's statement: Dormouse stole eggs (truth).- Dormouse stole eggs: but the one who stole eggs was lying, so Dormouse is lying.- Dormouse's statement: "I stole milk" is a lie, so he didn't steal milk.- Therefore, milk was stolen by March Hare.- March Hare's statement: "Hatter stole butter" is true, so March Hare is truthful.- March Hare stole milk: no contradiction here.Yes, this seems to work.3. Lastly, suppose the Dormouse stole the butter. Then, the Dormouse must be telling the truth. So, his statement "I stole the milk" would be true, meaning he stole both butter and milk? That doesn't make sense because each person stole one item. Therefore, the Dormouse cannot have stolen the butter.So, the only consistent scenario is:- Hatter stole butter: tells truth.- Hatter's statement: Dormouse stole eggs (truth).- Dormouse stole eggs: but since the one who stole eggs was lying, Dormouse is lying.- Dormouse's statement: "I stole milk" is a lie, so he didn't steal milk.- Therefore, milk was stolen by March Hare.- March Hare's statement: "Hatter stole butter" is true, so March Hare is truthful.This fits all the given conditions.

💡First, I need to evaluate the expression (4 + (-8) div (-4) - (-1)).I should follow the order of operations, which means performing division before addition and subtraction.Calculating the division part: (-8 div -4 = 2).Next, I address the subtraction of a negative number, which is equivalent to addition: (-(-1) = +1).Now, substitute these results back into the expression: (4 + 2 + 1).Finally, perform the addition from left to right: (4 + 2 = 6), and then (6 + 1 = 7).Therefore, the final result of the expression is (7).

💡Alright, so I have this problem where ( x + frac{1}{x} = 5 ), and I need to find ( T_7 = x^7 + frac{1}{x^7} ). Hmm, okay. I remember that for these kinds of problems, there's a recursive relationship or some pattern that can help find higher powers without having to compute each one step by step. Let me try to recall how that works.First, I know that ( T_1 = x + frac{1}{x} = 5 ). That's given. Now, I think ( T_2 ) can be found by squaring ( T_1 ). Let me check that:( T_1^2 = left( x + frac{1}{x} right)^2 = x^2 + 2 + frac{1}{x^2} ).So, ( T_2 = x^2 + frac{1}{x^2} = T_1^2 - 2 = 5^2 - 2 = 25 - 2 = 23 ). Okay, that works.Now, what about ( T_3 )? I think there's a formula for that too. Maybe it's related to multiplying ( T_1 ) and ( T_2 ). Let me see:( T_1 times T_2 = left( x + frac{1}{x} right) left( x^2 + frac{1}{x^2} right) = x^3 + frac{1}{x^3} + x + frac{1}{x} ).So, that's ( T_3 + T_1 ). Therefore, ( T_3 = T_1 times T_2 - T_1 = 5 times 23 - 5 = 115 - 5 = 110 ). Got it, ( T_3 = 110 ).Moving on to ( T_4 ). I think a similar approach can be used here. Maybe squaring ( T_2 ):( T_2^2 = left( x^2 + frac{1}{x^2} right)^2 = x^4 + 2 + frac{1}{x^4} ).So, ( T_4 = x^4 + frac{1}{x^4} = T_2^2 - 2 = 23^2 - 2 = 529 - 2 = 527 ). That seems right.Now, ( T_5 ). Hmm, how do I get ( T_5 )? Maybe by multiplying ( T_1 ) and ( T_4 ):( T_1 times T_4 = left( x + frac{1}{x} right) left( x^4 + frac{1}{x^4} right) = x^5 + frac{1}{x^5} + x^3 + frac{1}{x^3} ).So, that's ( T_5 + T_3 ). Therefore, ( T_5 = T_1 times T_4 - T_3 = 5 times 527 - 110 = 2635 - 110 = 2525 ). Okay, ( T_5 = 2525 ).Next, ( T_6 ). I think I can use ( T_3 times T_3 ) or maybe ( T_2 times T_4 ). Let me try both and see which one works.First, ( T_3 times T_3 = 110 times 110 = 12100 ). But I don't know if that's directly useful. Alternatively, ( T_2 times T_4 = 23 times 527 ). Let me calculate that:23 times 500 is 11,500, and 23 times 27 is 621, so total is 11,500 + 621 = 12,121. Hmm, that's the same as ( T_3^2 ). Wait, maybe that's not the right approach.Alternatively, maybe using ( T_1 times T_5 ):( T_1 times T_5 = 5 times 2525 = 12,625 ).But ( T_1 times T_5 = x^6 + frac{1}{x^6} + x^4 + frac{1}{x^4} ), which is ( T_6 + T_4 ).So, ( T_6 = T_1 times T_5 - T_4 = 12,625 - 527 = 12,098 ). Wait, is that right? Let me double-check:5 times 2525 is indeed 12,625. Then subtract 527: 12,625 - 500 = 12,125, then subtract 27 more: 12,125 - 27 = 12,098. Okay, so ( T_6 = 12,098 ).Now, finally, ( T_7 ). I think I can use ( T_1 times T_6 ) or ( T_2 times T_5 ) or ( T_3 times T_4 ). Let me try each and see.First, ( T_1 times T_6 = 5 times 12,098 = 60,490 ).But ( T_1 times T_6 = x^7 + frac{1}{x^7} + x^5 + frac{1}{x^5} = T_7 + T_5 ).So, ( T_7 = T_1 times T_6 - T_5 = 60,490 - 2525 = 57,965 ).Alternatively, let's check with ( T_2 times T_5 = 23 times 2525 ).23 times 2000 is 46,000, 23 times 500 is 11,500, and 23 times 25 is 575. So, total is 46,000 + 11,500 = 57,500 + 575 = 58,075.But ( T_2 times T_5 = x^7 + frac{1}{x^7} + x^3 + frac{1}{x^3} = T_7 + T_3 ).So, ( T_7 = T_2 times T_5 - T_3 = 58,075 - 110 = 57,965 ). Okay, same result.Just to be thorough, let's try ( T_3 times T_4 = 110 times 527 ).110 times 500 is 55,000, 110 times 27 is 2,970, so total is 55,000 + 2,970 = 57,970.But ( T_3 times T_4 = x^7 + frac{1}{x^7} + x + frac{1}{x} = T_7 + T_1 ).So, ( T_7 = T_3 times T_4 - T_1 = 57,970 - 5 = 57,965 ). Perfect, same answer again.So, all methods lead to ( T_7 = 57,965 ). Wait, but in the initial thought process, I thought of 57,960. Hmm, maybe I made a mistake earlier. Let me check my calculations again.Wait, when I calculated ( T_7 ) using ( T_3 times T_4 ), I got 57,970 - 5 = 57,965. But in the initial thought process, I thought it was 57,960. Maybe I miscalculated somewhere.Wait, let's go back to ( T_6 ). I had ( T_6 = 12,098 ). Then, ( T_7 = 5 times 12,098 - 2525 ).5 times 12,098: 12,000 x 5 = 60,000, 98 x 5 = 490, so total is 60,490. Then subtract 2525: 60,490 - 2,525 = 57,965.Yes, that's correct. So, the correct answer should be 57,965, not 57,960 as I initially thought. Maybe I made a typo or miscalculation earlier.Wait, let me check the initial thought process again. Oh, I see, in the initial thought process, the user had:( T_7 = 110 times 527 - 5 = 57,965 - 5 = 57,960 ). Wait, that's incorrect because 110 x 527 is 57,970, not 57,965. So, 57,970 - 5 = 57,965, not 57,960.So, the initial thought process had a mistake in the multiplication step. Therefore, the correct answer is 57,965.But wait, in my calculations above, I got 57,965 using all three methods. So, I think that's the correct answer.Wait, but let me double-check the multiplication of 110 x 527.110 x 500 = 55,000110 x 27 = 2,970Total = 55,000 + 2,970 = 57,970Yes, that's correct. So, 57,970 - 5 = 57,965.Therefore, the correct value of ( T_7 ) is 57,965.Wait, but in the initial thought process, the user had 57,960, which is incorrect. So, I think the correct answer is 57,965.But just to be absolutely sure, let me try another approach. Maybe using the recurrence relation.I recall that ( T_{m} = T_{m-1} times T_1 - T_{m-2} ).So, starting from ( T_1 = 5 ), ( T_2 = 23 ), ( T_3 = 110 ), ( T_4 = 527 ), ( T_5 = 2525 ), ( T_6 = 12,098 ), then ( T_7 = T_6 times T_1 - T_5 = 12,098 x 5 - 2525 = 60,490 - 2525 = 57,965 ).Yes, that confirms it again.So, I think the correct answer is 57,965, not 57,960 as initially thought. The mistake was in the initial thought process where 110 x 527 was miscalculated as 57,965 instead of 57,970, leading to an incorrect final result.

💡Alright, so I have this problem about an arithmetic sequence. Let me try to figure it out step by step. First, the problem says that in the arithmetic sequence {a_n}, the first term a₁ is 0, and the common difference d is not zero. Okay, so it's an arithmetic sequence starting at 0 with some common difference d. Then, it says that a_k equals the sum of the first seven terms, that is, a₁ + a₂ + a₃ + ... + a₇. I need to find the value of k. The options are 22, 23, 24, or 25. Alright, let me recall what an arithmetic sequence is. In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. So, the nth term can be expressed as a_n = a₁ + (n-1)d. Since a₁ is 0, that simplifies to a_n = (n-1)d. So, for example, a₂ would be (2-1)d = d, a₃ would be (3-1)d = 2d, and so on. Now, I need to find a_k, which is equal to the sum of the first seven terms. Let me write that down: a_k = a₁ + a₂ + a₃ + a₄ + a₅ + a₆ + a₇. Since a₁ is 0, that term is just 0. So, the sum becomes a₂ + a₃ + a₄ + a₅ + a₆ + a₇. Let me express each of these terms using the formula a_n = (n-1)d:- a₂ = (2-1)d = d- a₃ = (3-1)d = 2d- a₄ = (4-1)d = 3d- a₅ = (5-1)d = 4d- a₆ = (6-1)d = 5d- a₇ = (7-1)d = 6dSo, the sum is d + 2d + 3d + 4d + 5d + 6d. Let me add these up:d + 2d = 3d 3d + 3d = 6d 6d + 4d = 10d 10d + 5d = 15d 15d + 6d = 21dSo, the sum of the first seven terms is 21d. Therefore, a_k = 21d.But a_k is also equal to (k-1)d, as per the formula for the nth term of an arithmetic sequence. So, I can set up the equation:(k - 1)d = 21dSince d is not zero, I can divide both sides by d to get:k - 1 = 21Then, solving for k:k = 21 + 1 k = 22So, k is 22. Looking back at the options, A is 22. Therefore, the answer should be A.Wait, let me double-check my steps to make sure I didn't make a mistake. 1. I identified that a_n = (n-1)d because a₁ = 0.2. I calculated each term from a₂ to a₇ correctly: d, 2d, 3d, 4d, 5d, 6d.3. Summing them up: d + 2d + 3d + 4d + 5d + 6d = 21d. That seems right.4. Then, since a_k = 21d and a_k = (k-1)d, setting them equal gives (k-1)d = 21d.5. Dividing both sides by d (since d ≠ 0) gives k - 1 = 21, so k = 22.Everything seems to check out. I don't see any errors in my calculations. So, I'm confident that the answer is 22, which is option A.**Final Answer**boxed{A}

💡Okay, so I have this problem: Find the smallest natural number ( n ) for which ( n^{2} + 20n + 19 ) is divisible by 2019. Hmm, that seems a bit tricky, but let me try to break it down step by step.First, I know that if a number is divisible by 2019, it must be divisible by all the prime factors of 2019. Let me factorize 2019 to see what I'm dealing with. 2019 divided by 3 is 673, right? So, 2019 = 3 × 673. Both 3 and 673 are prime numbers, so that's the prime factorization. That means for ( n^{2} + 20n + 19 ) to be divisible by 2019, it must be divisible by both 3 and 673.Let me write the quadratic expression as a product of two binomials to see if that helps. So, ( n^{2} + 20n + 19 ). I need two numbers that multiply to 19 and add up to 20. Well, 1 and 19 do that. So, it factors to ( (n + 1)(n + 19) ).Alright, so ( (n + 1)(n + 19) ) needs to be divisible by 3 and 673. Let's tackle each prime factor separately.Starting with divisibility by 3: For a product to be divisible by 3, at least one of the factors must be divisible by 3. So, either ( n + 1 ) is divisible by 3 or ( n + 19 ) is divisible by 3. Let's see:If ( n + 1 equiv 0 mod 3 ), then ( n equiv -1 mod 3 ), which is the same as ( n equiv 2 mod 3 ).If ( n + 19 equiv 0 mod 3 ), then ( n equiv -19 mod 3 ). Since -19 divided by 3 is -6 with a remainder of 1, that's the same as ( n equiv 1 mod 3 ).So, ( n ) must be congruent to either 1 or 2 modulo 3. That gives me two possibilities for ( n ) in terms of modulo 3.Now, moving on to divisibility by 673. Since 673 is a prime number, similar logic applies. Either ( n + 1 ) is divisible by 673 or ( n + 19 ) is divisible by 673.Let's consider both cases:1. If ( n + 1 equiv 0 mod 673 ), then ( n equiv -1 mod 673 ), which is the same as ( n equiv 672 mod 673 ).2. If ( n + 19 equiv 0 mod 673 ), then ( n equiv -19 mod 673 ). Calculating -19 modulo 673, it's the same as 673 - 19 = 654, so ( n equiv 654 mod 673 ).So, ( n ) must be congruent to either 672 or 654 modulo 673.Now, I need to find the smallest natural number ( n ) that satisfies both the modulo 3 conditions and the modulo 673 conditions. This sounds like a problem where the Chinese Remainder Theorem (CRT) can be applied because 3 and 673 are coprime.Let me recall the Chinese Remainder Theorem. It states that if one has a system of simultaneous congruences with pairwise coprime moduli, then there exists a unique solution modulo the product of the moduli. In this case, the moduli are 3 and 673, which are coprime, so CRT applies.So, I have four possible combinations to consider because ( n ) can be either 1 or 2 mod 3 and either 654 or 672 mod 673. Let me list them:1. ( n equiv 1 mod 3 ) and ( n equiv 654 mod 673 )2. ( n equiv 1 mod 3 ) and ( n equiv 672 mod 673 )3. ( n equiv 2 mod 3 ) and ( n equiv 654 mod 673 )4. ( n equiv 2 mod 3 ) and ( n equiv 672 mod 673 )I need to solve each of these systems and find the smallest ( n ) that satisfies any of them.Let's start with the first combination:1. ( n equiv 1 mod 3 ) and ( n equiv 654 mod 673 )Let me express ( n ) as ( n = 673k + 654 ) for some integer ( k ). Now, substitute into the first congruence:( 673k + 654 equiv 1 mod 3 )Calculate ( 673 mod 3 ): 673 divided by 3 is 224 with a remainder of 1, so ( 673 equiv 1 mod 3 ).Similarly, ( 654 mod 3 ): 654 divided by 3 is exactly 218, so ( 654 equiv 0 mod 3 ).Therefore, the equation becomes:( 1 cdot k + 0 equiv 1 mod 3 )Simplify:( k equiv 1 mod 3 )So, ( k = 3m + 1 ) for some integer ( m ).Substitute back into ( n ):( n = 673(3m + 1) + 654 = 2019m + 673 + 654 = 2019m + 1327 )So, the solutions for this case are ( n = 2019m + 1327 ). The smallest natural number here is when ( m = 0 ), which gives ( n = 1327 ).Now, moving on to the second combination:2. ( n equiv 1 mod 3 ) and ( n equiv 672 mod 673 )Express ( n ) as ( n = 673k + 672 ).Substitute into the first congruence:( 673k + 672 equiv 1 mod 3 )Again, ( 673 equiv 1 mod 3 ) and ( 672 mod 3 ): 672 divided by 3 is 224, so ( 672 equiv 0 mod 3 ).Thus:( 1 cdot k + 0 equiv 1 mod 3 )Which simplifies to:( k equiv 1 mod 3 )So, ( k = 3m + 1 ).Substitute back into ( n ):( n = 673(3m + 1) + 672 = 2019m + 673 + 672 = 2019m + 1345 )The smallest ( n ) here is 1345 when ( m = 0 ).Third combination:3. ( n equiv 2 mod 3 ) and ( n equiv 654 mod 673 )Express ( n = 673k + 654 ).Substitute into the first congruence:( 673k + 654 equiv 2 mod 3 )Again, ( 673 equiv 1 mod 3 ) and ( 654 equiv 0 mod 3 ).So:( 1 cdot k + 0 equiv 2 mod 3 )Which gives:( k equiv 2 mod 3 )Thus, ( k = 3m + 2 ).Substitute back into ( n ):( n = 673(3m + 2) + 654 = 2019m + 1346 + 654 = 2019m + 2000 )The smallest ( n ) here is 2000 when ( m = 0 ).Fourth combination:4. ( n equiv 2 mod 3 ) and ( n equiv 672 mod 673 )Express ( n = 673k + 672 ).Substitute into the first congruence:( 673k + 672 equiv 2 mod 3 )Again, ( 673 equiv 1 mod 3 ) and ( 672 equiv 0 mod 3 ).So:( 1 cdot k + 0 equiv 2 mod 3 )Which gives:( k equiv 2 mod 3 )Thus, ( k = 3m + 2 ).Substitute back into ( n ):( n = 673(3m + 2) + 672 = 2019m + 1346 + 672 = 2019m + 2018 )The smallest ( n ) here is 2018 when ( m = 0 ).Now, compiling the smallest solutions from each case:1. 13272. 13453. 20004. 2018The smallest among these is 1327. Wait, but hold on, I need to verify if 1327 actually satisfies the original condition because sometimes when solving congruences, especially with multiple steps, it's easy to make a mistake.Let me compute ( n^2 + 20n + 19 ) for ( n = 1327 ) and see if it's divisible by 2019.First, compute ( n + 1 = 1328 ) and ( n + 19 = 1346 ). Then, multiply them: 1328 × 1346.But that's a big number. Alternatively, I can compute ( n^2 + 20n + 19 ) modulo 2019.Compute ( 1327^2 + 20×1327 + 19 mod 2019 ).But that's still a bit tedious. Maybe I can compute each term modulo 2019.First, compute ( 1327 mod 2019 ). It's 1327.Compute ( 1327^2 mod 2019 ). Let's compute 1327 squared:1327 × 1327. Let me compute 1300² = 1,690,000. Then, 27² = 729. Then, cross terms: 2×1300×27 = 70,200. So total is 1,690,000 + 70,200 + 729 = 1,760,929.Now, compute 1,760,929 mod 2019.To do this, divide 1,760,929 by 2019 and find the remainder.But that's still a bit involved. Maybe I can find a smarter way.Alternatively, note that 1327 = 2019 - 692, so 1327 ≡ -692 mod 2019.Therefore, 1327² ≡ (-692)² mod 2019.Compute 692²:692 × 692. Let's compute 700² = 490,000. Subtract 8×700 + 8²: 490,000 - 5600 - 64 = 490,000 - 5664 = 484,336.So, 692² = 484,336.Now, compute 484,336 mod 2019.Again, divide 484,336 by 2019.But perhaps we can find how many times 2019 goes into 484,336.Compute 2019 × 240 = 2019 × 200 + 2019 × 40 = 403,800 + 80,760 = 484,560.Wait, 2019 × 240 = 484,560, which is more than 484,336.So, 2019 × 239 = 484,560 - 2019 = 484,560 - 2,019 = 482,541.Now, 484,336 - 482,541 = 1,795.So, 484,336 ≡ 1,795 mod 2019.Therefore, 1327² ≡ 1,795 mod 2019.Next, compute 20 × 1327 mod 2019.20 × 1327 = 26,540.Now, compute 26,540 mod 2019.Divide 26,540 by 2019:2019 × 13 = 26,247.26,540 - 26,247 = 293.So, 20 × 1327 ≡ 293 mod 2019.Finally, add 19.So, total expression: 1,795 + 293 + 19 = 2,107.Now, compute 2,107 mod 2019: 2,107 - 2019 = 88.So, 2,107 ≡ 88 mod 2019.Wait, that's not zero. So, ( n = 1327 ) does not satisfy the condition. That means I made a mistake somewhere.Hmm, that's concerning. Maybe I made an error in my calculations. Let me double-check.First, when I computed ( 1327^2 mod 2019 ), I converted 1327 to -692 mod 2019, which is correct because 1327 + 692 = 2019.Then, I squared -692, which is 692² = 484,336.Then, I found 484,336 mod 2019 by subtracting 2019 × 239 = 482,541, which gives 484,336 - 482,541 = 1,795. So, 1327² ≡ 1,795 mod 2019.Then, 20 × 1327 = 26,540. Divided by 2019, 2019 × 13 = 26,247. So, 26,540 - 26,247 = 293. So, 20 × 1327 ≡ 293 mod 2019.Adding 19: 1,795 + 293 + 19 = 2,107. Then, 2,107 - 2019 = 88. So, 2,107 ≡ 88 mod 2019.So, the total is 88, not zero. Therefore, ( n = 1327 ) does not satisfy the condition. That means my earlier assumption that 1327 is a solution is incorrect.Hmm, that's a problem. Maybe I made a mistake in setting up the congruences.Let me go back to the beginning.We have ( (n + 1)(n + 19) equiv 0 mod 2019 ). So, either ( n + 1 equiv 0 mod 3 ) or ( n + 19 equiv 0 mod 3 ), and similarly, either ( n + 1 equiv 0 mod 673 ) or ( n + 19 equiv 0 mod 673 ).But perhaps I need to ensure that both conditions are satisfied simultaneously. That is, ( n + 1 ) or ( n + 19 ) is divisible by 3, and ( n + 1 ) or ( n + 19 ) is divisible by 673.But in my earlier approach, I considered all four combinations, but perhaps some of them don't actually satisfy the original equation because the divisibility by 3 and 673 might not align properly.Wait, but I used the Chinese Remainder Theorem, which should give valid solutions. Maybe I made a mistake in the calculations when verifying ( n = 1327 ).Alternatively, perhaps I should try another approach.Let me consider that ( (n + 1)(n + 19) equiv 0 mod 2019 ). So, either ( n + 1 equiv 0 mod 2019 ) or ( n + 19 equiv 0 mod 2019 ), or one factor is divisible by 3 and the other by 673.Wait, actually, since 2019 is 3 × 673, and 3 and 673 are coprime, the product ( (n + 1)(n + 19) ) must be divisible by both 3 and 673. So, either:- ( n + 1 equiv 0 mod 3 ) and ( n + 1 equiv 0 mod 673 ), or- ( n + 1 equiv 0 mod 3 ) and ( n + 19 equiv 0 mod 673 ), or- ( n + 19 equiv 0 mod 3 ) and ( n + 1 equiv 0 mod 673 ), or- ( n + 19 equiv 0 mod 3 ) and ( n + 19 equiv 0 mod 673 ).So, these are the four cases, which is what I considered earlier.But when I checked ( n = 1327 ), it didn't satisfy the condition. Maybe I need to check another solution.Looking back, the next smallest solution was 1345. Let me check ( n = 1345 ).Compute ( n + 1 = 1346 ) and ( n + 19 = 1364 ).Check if 1346 × 1364 is divisible by 2019.Alternatively, compute ( 1345^2 + 20×1345 + 19 mod 2019 ).Again, let's compute each term modulo 2019.First, ( n = 1345 ).Compute ( 1345 mod 2019 ). It's 1345.Compute ( 1345^2 mod 2019 ). Let's express 1345 as 2019 - 674, so 1345 ≡ -674 mod 2019.Thus, ( 1345^2 ≡ (-674)^2 = 674² mod 2019 ).Compute 674²:674 × 674. Let's compute 700² = 490,000. Subtract 26×700 + 26²: 490,000 - 18,200 - 676 = 490,000 - 18,876 = 471,124.Now, compute 471,124 mod 2019.Divide 471,124 by 2019:2019 × 233 = let's see, 2000 × 233 = 466,000, and 19 × 233 = 4,427. So, total is 466,000 + 4,427 = 470,427.Subtract from 471,124: 471,124 - 470,427 = 697.So, ( 1345^2 ≡ 697 mod 2019 ).Next, compute 20 × 1345 = 26,900.Compute 26,900 mod 2019.Divide 26,900 by 2019:2019 × 13 = 26,247.26,900 - 26,247 = 653.So, 20 × 1345 ≡ 653 mod 2019.Add 19: 697 + 653 + 19 = 1,369.Now, compute 1,369 mod 2019. It's 1,369, which is less than 2019, so it remains 1,369.So, ( n = 1345 ) gives a remainder of 1,369, not zero. So, that's not a solution either.Hmm, this is confusing. Maybe I need to check my approach again.Wait, perhaps I should consider that ( (n + 1)(n + 19) equiv 0 mod 2019 ) implies that either ( n + 1 equiv 0 mod 3 ) and ( n + 19 equiv 0 mod 673 ), or vice versa.So, let's consider the cases where one factor is divisible by 3 and the other by 673.Case 1: ( n + 1 equiv 0 mod 3 ) and ( n + 19 equiv 0 mod 673 ).So, ( n ≡ -1 mod 3 ) and ( n ≡ -19 mod 673 ).Which is ( n ≡ 2 mod 3 ) and ( n ≡ 654 mod 673 ).We already solved this earlier and got ( n = 2000 ) as the smallest solution.Case 2: ( n + 1 equiv 0 mod 673 ) and ( n + 19 equiv 0 mod 3 ).So, ( n ≡ -1 mod 673 ) and ( n ≡ -19 mod 3 ).Which is ( n ≡ 672 mod 673 ) and ( n ≡ 1 mod 3 ).We solved this and got ( n = 1345 ), but when we checked, it didn't work.Wait, maybe I made a mistake in the verification.Let me try ( n = 2000 ).Compute ( n + 1 = 2001 ) and ( n + 19 = 2019 ).So, ( 2001 × 2019 ). Clearly, 2019 is a factor, so the product is divisible by 2019. Therefore, ( n = 2000 ) is a valid solution.Similarly, ( n = 2018 ) would give ( n + 1 = 2019 ) and ( n + 19 = 2037 ). 2019 is a factor, so that's also divisible by 2019.But when I checked ( n = 1327 ) and ( n = 1345 ), they didn't work. So, perhaps only the solutions where both factors are aligned with the moduli work, i.e., when ( n + 1 ) is divisible by both 3 and 673, or ( n + 19 ) is divisible by both 3 and 673.Wait, but in the cases where one factor is divisible by 3 and the other by 673, the product should still be divisible by 2019 because 3 and 673 are coprime. So, why didn't ( n = 1345 ) work?Let me double-check ( n = 1345 ).Compute ( n + 1 = 1346 ) and ( n + 19 = 1364 ).Check if 1346 is divisible by 3: 1 + 3 + 4 + 6 = 14, which is not divisible by 3. So, 1346 is not divisible by 3.Check if 1364 is divisible by 3: 1 + 3 + 6 + 4 = 14, again not divisible by 3.Wait, that's a problem. If neither ( n + 1 ) nor ( n + 19 ) is divisible by 3, then their product isn't divisible by 3, hence not divisible by 2019.But according to our earlier setup, ( n = 1345 ) should satisfy ( n ≡ 1 mod 3 ) and ( n ≡ 672 mod 673 ). Let's check:1345 mod 3: 1 + 3 + 4 + 5 = 13, 1 + 3 = 4, 4 mod 3 = 1. So, 1345 ≡ 1 mod 3. Correct.1345 mod 673: 673 × 2 = 1346, so 1345 = 673 × 2 - 1 = 1346 - 1 = 1345. So, 1345 ≡ -1 mod 673, which is 672 mod 673. Correct.But then, why is neither ( n + 1 ) nor ( n + 19 ) divisible by 3?Wait, if ( n ≡ 1 mod 3 ), then ( n + 1 ≡ 2 mod 3 ) and ( n + 19 ≡ 1 + 19 = 20 ≡ 2 mod 3 ). So, neither is divisible by 3. That's the issue.So, my mistake was in assuming that if ( n ≡ 1 mod 3 ), then one of ( n + 1 ) or ( n + 19 ) would be divisible by 3, but in reality, neither is. Therefore, the product ( (n + 1)(n + 19) ) would not be divisible by 3, hence not by 2019.Wait, that contradicts my earlier reasoning. Let me think again.I thought that since 3 divides the product, it must divide at least one of the factors. But in this case, ( n ≡ 1 mod 3 ), so ( n + 1 ≡ 2 mod 3 ) and ( n + 19 ≡ 2 mod 3 ). So, neither is divisible by 3, meaning the product isn't divisible by 3. Therefore, ( n = 1345 ) is not a solution, which aligns with my earlier verification.So, my mistake was in the initial assumption that if ( n ≡ 1 mod 3 ), then one of ( n + 1 ) or ( n + 19 ) would be divisible by 3. But actually, since 19 ≡ 1 mod 3, adding 19 to ( n ) when ( n ≡ 1 mod 3 ) gives ( n + 19 ≡ 1 + 1 = 2 mod 3 ). So, both ( n + 1 ) and ( n + 19 ) are ≡ 2 mod 3, meaning neither is divisible by 3.Therefore, the only valid cases are when ( n + 1 ) is divisible by both 3 and 673, or ( n + 19 ) is divisible by both 3 and 673.So, let's reconsider the cases:Case 1: ( n + 1 equiv 0 mod 3 ) and ( n + 1 equiv 0 mod 673 ). So, ( n + 1 equiv 0 mod 2019 ). Therefore, ( n = 2019k - 1 ). The smallest natural number here is when ( k = 1 ), so ( n = 2018 ).Case 2: ( n + 19 equiv 0 mod 3 ) and ( n + 19 equiv 0 mod 673 ). So, ( n + 19 equiv 0 mod 2019 ). Therefore, ( n = 2019k - 19 ). The smallest natural number here is when ( k = 1 ), so ( n = 2019 - 19 = 2000 ).So, the two possible smallest solutions are 2000 and 2018. Between these, 2000 is smaller.Let me verify ( n = 2000 ):Compute ( n + 1 = 2001 ) and ( n + 19 = 2019 ).2001 is divisible by 3 (since 2 + 0 + 0 + 1 = 3) and 2019 is clearly divisible by 2019. Therefore, their product is divisible by 2019.Similarly, ( n = 2018 ):( n + 1 = 2019 ), which is divisible by 2019, and ( n + 19 = 2037 ), which is divisible by 3 (2 + 0 + 3 + 7 = 12). So, their product is divisible by 2019.Therefore, the smallest natural number ( n ) is 2000.

💡Alright, so I've got this hyperbola problem here. Let me try to figure it out step by step. The equation of the hyperbola is given as (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a > 0) and (b > 0). The foci of this hyperbola are (F_1) and (F_2), which are the left and right foci, respectively. There's a point (P) on the hyperbola such that ((|PF_1| - |PF_2|)^2 = b^2 - 3ab). I need to find the eccentricity of the hyperbola.First, I remember that for a hyperbola, the definition involves the difference of distances from any point on the hyperbola to the two foci. Specifically, for any point (P) on the hyperbola, the absolute difference of the distances to the foci is constant and equal to (2a). So, (|PF_1| - |PF_2| = 2a) or (|PF_2| - |PF_1| = 2a), depending on which branch of the hyperbola the point (P) is on.But in this problem, they've given me ((|PF_1| - |PF_2|)^2 = b^2 - 3ab). Hmm, that's interesting. So, squaring the difference of the distances gives me (b^2 - 3ab). I wonder if I can relate this to the standard properties of the hyperbola.Let me write down what I know:1. The hyperbola equation: (frac{x^2}{a^2} - frac{y^2}{b^2} = 1).2. The foci are located at ((pm c, 0)), where (c^2 = a^2 + b^2).3. The eccentricity (e) is given by (e = frac{c}{a}).4. For any point (P) on the hyperbola, (|PF_1| - |PF_2| = pm 2a).Given that, the problem states that ((|PF_1| - |PF_2|)^2 = b^2 - 3ab). Since (|PF_1| - |PF_2| = pm 2a), squaring both sides gives ((2a)^2 = 4a^2). So, according to the problem, (4a^2 = b^2 - 3ab).Wait, that seems a bit off. Let me double-check. If (|PF_1| - |PF_2| = 2a), then squaring both sides would indeed give (4a^2). So, the equation becomes:[4a^2 = b^2 - 3ab]Hmm, that's a quadratic equation in terms of (a) and (b). Maybe I can solve for one variable in terms of the other.Let me rearrange the equation:[4a^2 + 3ab - b^2 = 0]This looks like a quadratic in (a). Let me treat (b) as a constant and solve for (a). Let me write it as:[4a^2 + 3ab - b^2 = 0]Using the quadratic formula for (a):[a = frac{-3b pm sqrt{(3b)^2 - 4 cdot 4 cdot (-b^2)}}{2 cdot 4}]Calculating the discriminant:[(3b)^2 - 4 cdot 4 cdot (-b^2) = 9b^2 + 16b^2 = 25b^2]So, the square root of the discriminant is (5b). Plugging back into the formula:[a = frac{-3b pm 5b}{8}]This gives two solutions:1. (a = frac{-3b + 5b}{8} = frac{2b}{8} = frac{b}{4})2. (a = frac{-3b - 5b}{8} = frac{-8b}{8} = -b)But since (a > 0) and (b > 0), the second solution (a = -b) doesn't make sense because (a) must be positive. So, we discard that and take (a = frac{b}{4}).Alright, so (a = frac{b}{4}). Now, I need to find the eccentricity (e). To do that, I need to find (c), the distance from the center to each focus. For hyperbolas, (c^2 = a^2 + b^2).Let me compute (c):[c = sqrt{a^2 + b^2}]Substituting (a = frac{b}{4}):[c = sqrt{left(frac{b}{4}right)^2 + b^2} = sqrt{frac{b^2}{16} + b^2}]To combine the terms, I'll express (b^2) as (frac{16b^2}{16}):[c = sqrt{frac{b^2}{16} + frac{16b^2}{16}} = sqrt{frac{17b^2}{16}} = frac{sqrt{17}b}{4}]So, (c = frac{sqrt{17}b}{4}). Now, the eccentricity (e) is given by (e = frac{c}{a}).Substituting the values of (c) and (a):[e = frac{frac{sqrt{17}b}{4}}{frac{b}{4}} = sqrt{17}]Wait, that simplifies nicely. The (b) and the 4 cancel out, leaving just (sqrt{17}). So, the eccentricity is (sqrt{17}).Let me just verify my steps to make sure I didn't make any mistakes. I started with the given equation, recognized the hyperbola definition, set up the equation (4a^2 = b^2 - 3ab), solved for (a) in terms of (b), found (c) using (c^2 = a^2 + b^2), and then computed the eccentricity. Everything seems to check out.I think that's it. The eccentricity is (sqrt{17}), which corresponds to option D.**Final Answer**boxed{D}