answer：Okay, so I have this algebra problem here: I need to find the constant term in the expansion of (1 + x + a x³) multiplied by (x + 1/x)^5. They also told me that the sum of the coefficients of this expansion is 96, and I need to find the constant term. Hmm, okay, let me try to figure this out step by step.First, I remember that the sum of the coefficients in a polynomial is found by plugging in x = 1. So, if I substitute x = 1 into the entire expression, I should get 96. Let me write that down:Sum of coefficients = (1 + 1 + a * 1³) * (1 + 1/1)^5 = 96.Simplify that:(1 + 1 + a) * (2)^5 = 96.So, (2 + a) * 32 = 96.Wait, because 2^5 is 32. So, (2 + a) * 32 = 96.Let me solve for a:Divide both sides by 32: 2 + a = 96 / 32.96 divided by 32 is 3. So, 2 + a = 3.Subtract 2: a = 1.Okay, so a is 1. Got that.Now, I need to find the constant term in the expansion of (1 + x + x³) * (x + 1/x)^5.Hmm, so maybe I should first expand (x + 1/x)^5, and then multiply it by (1 + x + x³), and find the constant term.Alternatively, maybe I can find the constant term without expanding everything, by considering the contributions from each part.Let me think. The constant term comes from multiplying terms in (1 + x + x³) with terms in (x + 1/x)^5 such that the exponents add up to zero.So, if I denote (x + 1/x)^5 as a sum of terms with exponents from -5 to 5, then multiplying by 1, x, or x³ will shift the exponents accordingly.So, to get the constant term (exponent 0), I need:- The term from (x + 1/x)^5 with exponent 0 multiplied by 1.- The term from (x + 1/x)^5 with exponent -1 multiplied by x.- The term from (x + 1/x)^5 with exponent -3 multiplied by x³.So, if I can find the coefficients of x^0, x^{-1}, and x^{-3} in (x + 1/x)^5, then multiply them by 1, x, and x³ respectively, and sum them up, that should give me the constant term.Alternatively, since (x + 1/x)^5 is symmetric, the coefficients of x^k and x^{-k} are the same.So, let me recall the expansion of (x + 1/x)^5.Using the binomial theorem, (x + 1/x)^5 = Σ_{k=0}^5 C(5, k) x^{5 - 2k}.So, the exponents are 5, 3, 1, -1, -3, -5.Therefore, the coefficients for exponents 0, -1, and -3 are:Wait, hold on, in the expansion, the exponents go from 5 down to -5, stepping by 2 each time. So, exponents are 5, 3, 1, -1, -3, -5.Therefore, the coefficient of x^0 is zero because there is no x^0 term in (x + 1/x)^5. Hmm, interesting.Wait, so does that mean the coefficient of x^0 is zero? Let me check.Wait, the expansion is:C(5,0)x^5 + C(5,1)x^3 + C(5,2)x^1 + C(5,3)x^{-1} + C(5,4)x^{-3} + C(5,5)x^{-5}.So, yeah, there is no x^0 term. So, the coefficient of x^0 is zero.Similarly, the coefficient of x^{-1} is C(5,3) = 10, and the coefficient of x^{-3} is C(5,4) = 5.So, going back, the constant term in the entire expansion is:(1 * coefficient of x^0) + (x * coefficient of x^{-1}) + (x³ * coefficient of x^{-3}).But since coefficient of x^0 is zero, that term is zero.So, the constant term is:0 + (1 * coefficient of x^{-1}) + (1 * coefficient of x^{-3}).Wait, hold on, no. Wait, when we multiply by x, we have to consider the coefficient of x^{-1} in (x + 1/x)^5, which is 10, and multiply it by x, so it becomes x * x^{-1} = x^0, which is 1. Similarly, x³ multiplied by x^{-3} is x^0, which is 1.Wait, maybe I need to think in terms of multiplying the polynomials.So, (1 + x + x³) * (x + 1/x)^5.Each term in (1 + x + x³) multiplies each term in (x + 1/x)^5.So, the constant term is the sum of:1 * (coefficient of x^0 in (x + 1/x)^5) +x * (coefficient of x^{-1} in (x + 1/x)^5) +x³ * (coefficient of x^{-3} in (x + 1/x)^5).But as we saw, (x + 1/x)^5 doesn't have an x^0 term, so the first term is zero.The second term is x multiplied by the coefficient of x^{-1}, which is 10, so that gives 10 * x^{0}.The third term is x³ multiplied by the coefficient of x^{-3}, which is 5, so that gives 5 * x^{0}.Therefore, the constant term is 0 + 10 + 5 = 15.Wait, so is it 15? Hmm, let me double-check.Alternatively, maybe I can compute the expansion of (x + 1/x)^5 first.Let me compute that:(x + 1/x)^5 = C(5,0)x^5 + C(5,1)x^3 + C(5,2)x + C(5,3)x^{-1} + C(5,4)x^{-3} + C(5,5)x^{-5}.Calculating the coefficients:C(5,0) = 1,C(5,1) = 5,C(5,2) = 10,C(5,3) = 10,C(5,4) = 5,C(5,5) = 1.So, (x + 1/x)^5 = x^5 + 5x^3 + 10x + 10/x + 5/x^3 + 1/x^5.So, written out:x^5 + 5x^3 + 10x + 10x^{-1} + 5x^{-3} + x^{-5}.Now, multiply this by (1 + x + x³):So, we have to compute:(1 + x + x³) * (x^5 + 5x^3 + 10x + 10x^{-1} + 5x^{-3} + x^{-5}).To find the constant term, we need the terms where the exponents add up to zero.So, let's consider each term in (1 + x + x³) multiplied by each term in the expansion above.1 * (x^5 + 5x^3 + 10x + 10x^{-1} + 5x^{-3} + x^{-5}):Looking for exponents that sum to zero when multiplied by 1 (which is x^0). So, we need terms with exponent 0 in this product. But in the expansion above, the exponents are 5, 3, 1, -1, -3, -5. So, multiplying by 1, the exponents remain the same. So, there is no x^0 term here.x * (x^5 + 5x^3 + 10x + 10x^{-1} + 5x^{-3} + x^{-5}):Multiplying each term by x, so exponents increase by 1:x^6 + 5x^4 + 10x^2 + 10x^0 + 5x^{-2} + x^{-4}.So, here, the constant term is 10.x³ * (x^5 + 5x^3 + 10x + 10x^{-1} + 5x^{-3} + x^{-5}):Multiplying each term by x³, exponents increase by 3:x^8 + 5x^6 + 10x^4 + 10x^2 + 5x^0 + x^{-2}.So, here, the constant term is 5.Therefore, adding up the constant terms from each part:From 1 * expansion: 0,From x * expansion: 10,From x³ * expansion: 5.Total constant term: 0 + 10 + 5 = 15.So, the constant term is 15.Wait, that seems consistent with my earlier calculation. So, I think 15 is the answer.But just to make sure, let me think if there's another way to approach this.Alternatively, I can consider that the constant term is the sum of the products of the coefficients of x^k in (1 + x + x³) and x^{-k} in (x + 1/x)^5.So, for k = 0: coefficient of x^0 in (1 + x + x³) is 1, and coefficient of x^0 in (x + 1/x)^5 is 0. So, 1*0 = 0.For k = 1: coefficient of x^1 in (1 + x + x³) is 1, and coefficient of x^{-1} in (x + 1/x)^5 is 10. So, 1*10 = 10.For k = 3: coefficient of x^3 in (1 + x + x³) is 1, and coefficient of x^{-3} in (x + 1/x)^5 is 5. So, 1*5 = 5.Adding these up: 0 + 10 + 5 = 15.Same result. Okay, so that seems solid.Therefore, the constant term is 15.**Final Answer**The constant term is boxed{15}.

answer：Okay, so I have this problem here where I'm given that ( a^m = 2 ) and ( a^n = 3 ). I need to find the value of ( a^{3m + 2n} ). The options are A: 24, B: 36, C: 72, D: 6. Hmm, let me think about how to approach this.First, I remember that when dealing with exponents, there are certain properties that can help simplify expressions. One of them is that ( a^{b + c} = a^b times a^c ). So, if I have an exponent that's a sum, I can break it down into the product of two separate exponents. Similarly, another property is that ( (a^b)^c = a^{b times c} ), which means if I have an exponent raised to another exponent, I can multiply them.Looking at the expression I need to find, ( a^{3m + 2n} ), I can try to break it down using these properties. Let me write it out step by step.First, I can separate the exponent into two parts: ( 3m ) and ( 2n ). So, ( a^{3m + 2n} = a^{3m} times a^{2n} ). That seems right because of the addition in the exponent.Now, I need to figure out what ( a^{3m} ) and ( a^{2n} ) are individually. Let's start with ( a^{3m} ). Using the second property I mentioned, ( (a^m)^3 = a^{m times 3} = a^{3m} ). So, ( a^{3m} ) is just ( (a^m)^3 ).But wait, I know that ( a^m = 2 ). So, substituting that in, ( (a^m)^3 = 2^3 ). Calculating that, ( 2^3 = 8 ). So, ( a^{3m} = 8 ).Alright, moving on to ( a^{2n} ). Similarly, using the same exponent rule, ( (a^n)^2 = a^{n times 2} = a^{2n} ). So, ( a^{2n} ) is ( (a^n)^2 ).Given that ( a^n = 3 ), substituting that in, ( (a^n)^2 = 3^2 ). Calculating that, ( 3^2 = 9 ). So, ( a^{2n} = 9 ).Now, going back to the original expression, ( a^{3m + 2n} = a^{3m} times a^{2n} = 8 times 9 ). Let me compute that: 8 multiplied by 9 is 72.Hmm, 72 is one of the options. Looking back at the choices, option C is 72. So, that must be the answer.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the beginning:1. ( a^m = 2 ) and ( a^n = 3 ).2. ( a^{3m + 2n} ) can be broken down into ( a^{3m} times a^{2n} ).3. ( a^{3m} = (a^m)^3 = 2^3 = 8 ).4. ( a^{2n} = (a^n)^2 = 3^2 = 9 ).5. Multiplying those together: 8 * 9 = 72.Yes, that all seems correct. I don't see any errors in my reasoning. So, I feel confident that the answer is 72, which is option C.**Final Answer**boxed{C}

answer：Okay, so I have this problem here: Given that the complex number (a + 2i)/(1 - i) is a pure imaginary number, where i is the imaginary unit, find the value of the real number a. Hmm, okay. Let me try to figure this out step by step.First, I remember that a complex number is pure imaginary if its real part is zero. So, if I can express (a + 2i)/(1 - i) in the standard form of a complex number, which is x + yi, where x is the real part and y is the imaginary part, then I can set the real part equal to zero and solve for a.But how do I convert that fraction into the standard form? I think I need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of 1 - i is 1 + i, right? So, if I multiply both the numerator and the denominator by 1 + i, that should help me eliminate the imaginary unit from the denominator.Let me write that down:(a + 2i)/(1 - i) * (1 + i)/(1 + i) = [(a + 2i)(1 + i)] / [(1 - i)(1 + i)]Okay, so now I need to multiply out the numerator and the denominator.Starting with the numerator: (a + 2i)(1 + i). Let me expand this using the distributive property.First, multiply a by 1, which is a. Then, a times i, which is ai. Next, 2i times 1, which is 2i. Finally, 2i times i, which is 2i². So, putting it all together:Numerator = a + ai + 2i + 2i²I know that i² is equal to -1, so 2i² is 2*(-1) = -2. So, substituting that in:Numerator = a + ai + 2i - 2Now, let me combine like terms. The real parts are a and -2, and the imaginary parts are ai and 2i.So, real part: a - 2Imaginary part: (a + 2)iSo, numerator simplifies to (a - 2) + (a + 2)i.Now, let's work on the denominator: (1 - i)(1 + i). This is a difference of squares, so it should be 1² - (i)².Calculating that:Denominator = 1 - i²Again, since i² = -1, this becomes:Denominator = 1 - (-1) = 1 + 1 = 2So, the denominator simplifies to 2.Putting it all together, the original expression becomes:[(a - 2) + (a + 2)i] / 2Which can be written as:(a - 2)/2 + (a + 2)/2 * iSo, now the complex number is expressed as ( (a - 2)/2 ) + ( (a + 2)/2 )i.Since the problem states that this complex number is pure imaginary, the real part must be zero. That means:(a - 2)/2 = 0Solving for a:Multiply both sides by 2:a - 2 = 0Add 2 to both sides:a = 2Wait, let me double-check my steps to make sure I didn't make a mistake.Starting with (a + 2i)/(1 - i). Multiply numerator and denominator by (1 + i):Numerator: (a + 2i)(1 + i) = a + ai + 2i + 2i² = a + ai + 2i - 2Yes, that's correct. Then, denominator: (1 - i)(1 + i) = 1 - i² = 1 - (-1) = 2. Correct.So, numerator is (a - 2) + (a + 2)i, denominator is 2. So, the expression is (a - 2)/2 + (a + 2)/2 i.Since it's pure imaginary, real part is zero: (a - 2)/2 = 0, so a = 2. Seems straightforward.But just to be thorough, let me plug a = 2 back into the original expression and see if it indeed results in a pure imaginary number.Original expression: (2 + 2i)/(1 - i)Multiply numerator and denominator by (1 + i):Numerator: (2 + 2i)(1 + i) = 2*1 + 2*i + 2i*1 + 2i*i = 2 + 2i + 2i + 2i²Simplify: 2 + 4i + 2*(-1) = 2 + 4i - 2 = 0 + 4iDenominator: (1 - i)(1 + i) = 1 - i² = 1 - (-1) = 2So, the expression becomes (0 + 4i)/2 = 0 + 2i, which is indeed a pure imaginary number.Therefore, a = 2 is correct.I think that's solid. I don't see any mistakes in my calculations or reasoning. So, the value of a is 2.**Final Answer**The value of the real number ( a ) is boxed{2}.

answer：To determine how many multiples of 3 are between 62 and 215, I'll start by identifying the smallest multiple of 3 that is greater than or equal to 62. Dividing 62 by 3 gives approximately 20.666, so the next whole number is 21. Multiplying 21 by 3 results in 63, which is the first multiple of 3 in the range.Next, I'll find the largest multiple of 3 that is less than or equal to 215. Dividing 215 by 3 gives approximately 71.666, so the whole number part is 71. Multiplying 71 by 3 gives 213, which is the last multiple of 3 in the range.Now, I'll calculate the total number of multiples of 3 between 63 and 213. This can be done by subtracting the smaller multiple's position (21) from the larger multiple's position (71) and adding 1 to include both endpoints. So, 71 minus 21 equals 50, and adding 1 gives a total of 51 multiples of 3.

answer：First, I need to determine the total number of faces on the cube. A standard cube has 6 faces.Next, I identify the number of blue faces, which is 4, and the number of red faces, which is 2.To find the probability of rolling a blue face, I divide the number of blue faces by the total number of faces. So, the probability is 4 divided by 6, which simplifies to 2/3.

answer：Alright, so I have this problem here: I need to find the maximum positive integer ( n ) such that ( n^2 leq 160 times 170 times 180 times 190 ). Hmm, okay. Let me try to figure this out step by step.First, I think I need to calculate the product of those four numbers: 160, 170, 180, and 190. That seems like a big number, so maybe I can find a smarter way to compute it without multiplying them all directly. Alternatively, I could compute it step by step.Let me write down the numbers: 160, 170, 180, 190. I notice that these are all multiples of 10. Maybe I can factor out the 10s first to simplify the multiplication. So, 160 is 16 × 10, 170 is 17 × 10, 180 is 18 × 10, and 190 is 19 × 10. So, if I factor out the 10s, I get:( 160 times 170 times 180 times 190 = (16 times 10) times (17 times 10) times (18 times 10) times (19 times 10) )That simplifies to:( (16 times 17 times 18 times 19) times (10 times 10 times 10 times 10) )Which is:( (16 times 17 times 18 times 19) times 10^4 )Okay, so now I just need to compute 16 × 17 × 18 × 19 and then multiply that by 10,000. Let me compute 16 × 17 first.16 × 17: Hmm, 16 × 10 is 160, and 16 × 7 is 112, so 160 + 112 = 272. So, 16 × 17 = 272.Now, I need to multiply that by 18. So, 272 × 18. Let me break that down. 272 × 10 is 2720, and 272 × 8 is 2176. Adding those together: 2720 + 2176 = 4896. So, 272 × 18 = 4896.Next, I need to multiply 4896 by 19. Hmm, 4896 × 19. Let me do 4896 × 20 first, which is 97,920, and then subtract 4896 to get 4896 × 19. So, 97,920 - 4,896 = 93,024. Let me check that subtraction: 97,920 minus 4,000 is 93,920, and then minus 896 is 93,024. Yeah, that seems right.So, putting it all together, 16 × 17 × 18 × 19 = 93,024. Therefore, the original product is 93,024 × 10,000, which is 930,240,000.Wait, let me make sure I did that correctly. 93,024 × 10,000 is indeed 930,240,000 because you just add four zeros. Okay, so that part seems correct.So, now the problem is to find the maximum integer ( n ) such that ( n^2 leq 930,240,000 ). That means I need to find the square root of 930,240,000 and take the floor of that value, right?Calculating the square root of 930,240,000. Hmm, that's a big number. Maybe I can simplify it by expressing it in terms of factors that are perfect squares.First, let me note that 930,240,000 is equal to 93,024 × 10,000. Since 10,000 is a perfect square (100^2), I can take that out of the square root.So, ( sqrt{930,240,000} = sqrt{93,024 times 10,000} = sqrt{93,024} times sqrt{10,000} = sqrt{93,024} times 100 ).Therefore, I just need to compute ( sqrt{93,024} ) and then multiply by 100.Now, let's compute ( sqrt{93,024} ). Hmm, 93,024. Let me see if I can factor this number to find perfect squares.First, let's note that 93,024 ÷ 16 is 5,814 because 16 × 5,814 = 93,024. So, 93,024 = 16 × 5,814.So, ( sqrt{93,024} = sqrt{16 times 5,814} = sqrt{16} times sqrt{5,814} = 4 times sqrt{5,814} ).Now, I need to compute ( sqrt{5,814} ). Hmm, 5,814. Let me see if I can factor this further.5,814 ÷ 2 = 2,907. So, 5,814 = 2 × 2,907.2,907 ÷ 3 = 969. So, 2,907 = 3 × 969.969 ÷ 3 = 323. So, 969 = 3 × 323.323 is a prime number? Let me check. 323 ÷ 17 = 19, because 17 × 19 = 323. Yes, that's right.So, putting it all together, 5,814 factors into 2 × 3 × 3 × 17 × 19. So, 5,814 = 2 × 3² × 17 × 19.Therefore, ( sqrt{5,814} = sqrt{2 times 3^2 times 17 times 19} = 3 times sqrt{2 times 17 times 19} ).Calculating the inside of the square root: 2 × 17 × 19. 2 × 17 is 34, and 34 × 19 is 646. So, ( sqrt{5,814} = 3 times sqrt{646} ).So, now, ( sqrt{93,024} = 4 times 3 times sqrt{646} = 12 times sqrt{646} ).Therefore, ( sqrt{930,240,000} = 12 times sqrt{646} times 100 = 1,200 times sqrt{646} ).Hmm, okay, so now I need to compute ( sqrt{646} ). Let's see, 646 is between 25²=625 and 26²=676. So, sqrt(646) is between 25 and 26.Let me compute 25.4²: 25 × 25 = 625, 0.4² = 0.16, and the cross term is 2 × 25 × 0.4 = 20. So, 25.4² = 625 + 20 + 0.16 = 645.16. Oh, that's really close to 646.So, 25.4² = 645.16, which is just 0.84 less than 646. So, sqrt(646) is approximately 25.4 + (0.84)/(2×25.4) ≈ 25.4 + 0.0165 ≈ 25.4165.So, sqrt(646) ≈ 25.4165.Therefore, sqrt(930,240,000) ≈ 1,200 × 25.4165 ≈ 1,200 × 25.4165.Let me compute that: 1,200 × 25 = 30,000, and 1,200 × 0.4165 = 1,200 × 0.4 = 480, plus 1,200 × 0.0165 = 19.8. So, 480 + 19.8 = 499.8. Therefore, total is 30,000 + 499.8 = 30,499.8.So, sqrt(930,240,000) ≈ 30,499.8. Therefore, the maximum integer ( n ) such that ( n^2 leq 930,240,000 ) is 30,499 because 30,499² is less than or equal to 930,240,000, and 30,500² would be more.Wait, but let me verify this because my approximation might be off. Maybe I should compute 30,500² and see if it's more than 930,240,000.30,500² is (30,000 + 500)² = 30,000² + 2×30,000×500 + 500² = 900,000,000 + 30,000,000 + 250,000 = 930,250,000.Wait, that's interesting. 30,500² is 930,250,000, which is actually 10,000 more than 930,240,000. So, 30,500² is larger than our target number.Therefore, 30,500 is too big. So, the maximum integer ( n ) must be 30,499.But wait, let me compute 30,499² to make sure.30,499²: Let's compute (30,500 - 1)² = 30,500² - 2×30,500×1 + 1² = 930,250,000 - 61,000 + 1 = 930,250,000 - 61,000 is 929,189,000, plus 1 is 929,189,001.Wait, that can't be right because 30,500² is 930,250,000, so subtracting 61,000 gives 929,189,000, which is way less than 930,240,000. That doesn't make sense because 30,499 should be just slightly less than 30,500, so 30,499² should be just slightly less than 30,500², which is 930,250,000.Wait, maybe I made a mistake in the calculation.Wait, let me compute 30,499² correctly.Let me use the formula (a - b)² = a² - 2ab + b², where a = 30,500 and b = 1.So, 30,499² = (30,500 - 1)² = 30,500² - 2×30,500×1 + 1² = 930,250,000 - 61,000 + 1 = 930,250,000 - 61,000 is 929,189,000, plus 1 is 929,189,001.Wait, that's way less than 930,240,000. That can't be right because 30,499 is just 1 less than 30,500, so the square should be just slightly less.Wait, maybe my initial assumption is wrong. Let me compute 30,499² another way.Alternatively, let me compute 30,499 × 30,499.Let me write it as (30,000 + 499)² = 30,000² + 2×30,000×499 + 499².Compute each term:30,000² = 900,000,000.2×30,000×499 = 60,000×499. Let's compute 60,000×500 = 30,000,000, subtract 60,000 to get 60,000×499 = 29,940,000.499²: Let's compute 500² = 250,000, subtract 2×500 + 1 = 1000 + 1 = 1001, so 250,000 - 1001 = 249,999.So, adding all together: 900,000,000 + 29,940,000 = 929,940,000. Then, add 249,999: 929,940,000 + 249,999 = 930,189,999.So, 30,499² = 930,189,999.Wait, but our target is 930,240,000. So, 930,189,999 is less than 930,240,000. Therefore, 30,499² is less than the product, so 30,499 is a valid n. But is 30,500 too big? Yes, because 30,500² is 930,250,000, which is greater than 930,240,000.Therefore, the maximum integer n is 30,499.Wait, but earlier, when I approximated sqrt(930,240,000) as 30,499.8, which is very close to 30,500. So, that makes sense because 30,499.8 is just slightly less than 30,500, so 30,499 is the integer part.But just to make sure, let me compute 30,499² and 30,500² again.30,499² = 930,189,999.30,500² = 930,250,000.Our target is 930,240,000. So, 30,499² is 930,189,999, which is less than 930,240,000. The difference is 930,240,000 - 930,189,999 = 50,001.So, 30,499² is 50,001 less than the target. So, is there a number between 30,499 and 30,500 whose square is exactly 930,240,000? No, because n has to be an integer. So, 30,499 is the largest integer where n² is less than or equal to 930,240,000.Wait, but hold on a second. Let me check my earlier calculation of the product. I had 160 × 170 × 180 × 190 = 930,240,000. Let me verify that because that seems crucial.160 × 170: 160 × 170. Let's compute 16 × 17 = 272, so 160 × 170 = 27,200.180 × 190: 18 × 19 = 342, so 180 × 190 = 34,200.Now, multiply 27,200 × 34,200. Hmm, that's a big number, but let me compute it step by step.27,200 × 34,200 = (27,200 × 34) × 1,000.Compute 27,200 × 34:27,200 × 30 = 816,000.27,200 × 4 = 108,800.Adding together: 816,000 + 108,800 = 924,800.Therefore, 27,200 × 34,200 = 924,800 × 1,000 = 924,800,000.Wait, that's different from what I had earlier. Earlier, I had 930,240,000, but now I have 924,800,000. So, which one is correct?Wait, let me recast the original multiplication.160 × 170 × 180 × 190.Alternatively, group them as (160 × 190) × (170 × 180).Compute 160 × 190: 16 × 19 = 304, so 160 × 190 = 30,400.Compute 170 × 180: 17 × 18 = 306, so 170 × 180 = 30,600.Now, multiply 30,400 × 30,600.Compute 30,400 × 30,600 = (30,400 × 30) × 1,000 + (30,400 × 60) × 1,000.Wait, no, that's not the right way. Let me compute 30,400 × 30,600.Alternatively, 30,400 × 30,600 = (30,000 + 400) × (30,000 + 600).Using the formula (a + b)(c + d) = ac + ad + bc + bd.So, 30,000 × 30,000 = 900,000,000.30,000 × 600 = 18,000,000.400 × 30,000 = 12,000,000.400 × 600 = 240,000.Adding them all together:900,000,000 + 18,000,000 = 918,000,000.918,000,000 + 12,000,000 = 930,000,000.930,000,000 + 240,000 = 930,240,000.Ah, okay, so that's correct. So, the product is indeed 930,240,000. So, my initial calculation was correct, and the second way I computed it was wrong because I grouped them incorrectly.So, 160 × 170 × 180 × 190 is indeed 930,240,000.Therefore, going back, sqrt(930,240,000) is approximately 30,499.8, so the maximum integer n is 30,499.But just to make sure, let me compute 30,499² and 30,500² again.30,499²: As computed earlier, it's 930,189,999.30,500²: 930,250,000.Our target is 930,240,000.So, 30,499² = 930,189,999 < 930,240,000.30,500² = 930,250,000 > 930,240,000.Therefore, 30,499 is the largest integer n such that n² ≤ 930,240,000.But wait, let me check if there's a number between 30,499 and 30,500, say 30,499.5, whose square is exactly 930,240,000. But since n has to be an integer, we can't have a fractional n. So, 30,499 is indeed the maximum integer.Alternatively, maybe I can compute the exact square root of 930,240,000 and see how close it is to 30,500.Earlier, I had sqrt(930,240,000) = 1,200 × sqrt(646) ≈ 1,200 × 25.4165 ≈ 30,499.8.So, sqrt(930,240,000) ≈ 30,499.8, which is very close to 30,500, but still less. So, the integer part is 30,499.Therefore, the maximum integer n is 30,499.But just to be thorough, let me compute 30,499² and 30,500² once more.30,499 × 30,499:Let me compute 30,499 × 30,499.I can write this as (30,500 - 1) × (30,500 - 1) = 30,500² - 2×30,500×1 + 1².Which is 930,250,000 - 61,000 + 1 = 930,250,000 - 61,000 = 929,189,000 + 1 = 929,189,001.Wait, that's different from my earlier calculation. Wait, no, that can't be. Wait, 30,500² is 930,250,000, so subtracting 61,000 gives 929,189,000, which is way less than 930,240,000. That doesn't make sense.Wait, hold on, I think I made a mistake in the calculation.Wait, (a - b)² = a² - 2ab + b². So, (30,500 - 1)² = 30,500² - 2×30,500×1 + 1² = 930,250,000 - 61,000 + 1 = 930,250,000 - 61,000 is 929,189,000, plus 1 is 929,189,001.But that can't be, because 30,499 is just 1 less than 30,500, so 30,499² should be just slightly less than 30,500², which is 930,250,000.But 929,189,001 is way less. That must be a miscalculation.Wait, no, actually, 30,500² is 930,250,000, so 30,499² should be 930,250,000 - 2×30,500 + 1 = 930,250,000 - 61,000 + 1 = 930,189,001.Wait, that's still 930,189,001, which is 50,999 less than 930,240,000.Wait, that can't be right because 30,499 is just 1 less than 30,500, so the squares should be very close.Wait, maybe I'm making a mistake in the arithmetic.Wait, 30,500² is 930,250,000.30,499² = (30,500 - 1)² = 30,500² - 2×30,500×1 + 1² = 930,250,000 - 61,000 + 1 = 930,250,000 - 61,000 is 929,189,000, plus 1 is 929,189,001.Wait, that's correct, but 929,189,001 is way less than 930,240,000. That seems contradictory because 30,499 is just 1 less than 30,500, so their squares should be close.Wait, actually, no. Wait, 30,500 is 30,500, so 30,499 is 1 less, but the square of 30,500 is 930,250,000, so the square of 30,499 is 930,250,000 minus 2×30,500 + 1, which is 930,250,000 - 61,000 + 1 = 929,189,001.Wait, that's correct, but that's a difference of 61,000 - 1, which is 60,999. So, 30,499² is 60,999 less than 30,500².But 30,500² is 930,250,000, so 30,499² is 930,250,000 - 60,999 = 930,189,001.Wait, but 930,189,001 is still less than 930,240,000.So, the difference between 930,240,000 and 930,189,001 is 50,999.So, 30,499² = 930,189,001.30,500² = 930,250,000.Our target is 930,240,000, which is between these two.Therefore, 30,499² < 930,240,000 < 30,500².So, the maximum integer n is 30,499.Wait, but earlier, when I approximated sqrt(930,240,000) as 30,499.8, which is just 0.2 less than 30,500, so 30,499.8 is approximately the square root, so the integer part is 30,499.Therefore, the maximum integer n is 30,499.But just to make sure, let me compute 30,499.8² and see if it's approximately 930,240,000.Compute 30,499.8²:Let me write it as (30,500 - 0.2)² = 30,500² - 2×30,500×0.2 + (0.2)² = 930,250,000 - 12,200 + 0.04 = 930,250,000 - 12,200 = 929,238,000 + 0.04 = 929,238,000.04.Wait, that's not right because 30,499.8 is less than 30,500, so its square should be less than 30,500², which is 930,250,000.But 929,238,000 is way less than 930,240,000. That can't be right.Wait, no, I think I made a mistake in the calculation.Wait, 30,500 - 0.2 is 30,499.8.So, (30,500 - 0.2)² = 30,500² - 2×30,500×0.2 + (0.2)² = 930,250,000 - 12,200 + 0.04 = 930,250,000 - 12,200 = 929,238,000 + 0.04 = 929,238,000.04.Wait, that's correct, but that's way less than 930,240,000. So, that can't be right because 30,499.8 is supposed to be the square root of 930,240,000.Wait, no, that can't be. There must be a mistake in my approximation.Wait, earlier, I had sqrt(930,240,000) = 1,200 × sqrt(646) ≈ 1,200 × 25.4165 ≈ 30,499.8.But when I compute (30,499.8)², I get 929,238,000.04, which is way less than 930,240,000.That means my approximation was wrong.Wait, that can't be. There must be a miscalculation.Wait, let me compute sqrt(646) again.Earlier, I thought sqrt(646) ≈ 25.4165 because 25.4² = 645.16, which is close to 646.But let's compute 25.4²: 25 × 25 = 625, 0.4² = 0.16, and 2×25×0.4 = 20. So, 25.4² = 625 + 20 + 0.16 = 645.16.So, 25.4² = 645.16.So, 646 - 645.16 = 0.84.So, to find sqrt(646), we can use linear approximation.Let me denote f(x) = sqrt(x). We know f(645.16) = 25.4.We want f(646). The derivative f’(x) = 1/(2sqrt(x)).So, f(646) ≈ f(645.16) + f’(645.16) × (646 - 645.16).Which is 25.4 + (1/(2×25.4)) × 0.84.Compute 1/(2×25.4) = 1/50.8 ≈ 0.019685.Multiply by 0.84: 0.019685 × 0.84 ≈ 0.01647.So, sqrt(646) ≈ 25.4 + 0.01647 ≈ 25.41647.So, sqrt(646) ≈ 25.4165.Therefore, sqrt(930,240,000) = 1,200 × 25.4165 ≈ 1,200 × 25.4165.Compute 1,200 × 25 = 30,000.1,200 × 0.4165 = 1,200 × 0.4 = 480, and 1,200 × 0.0165 = 19.8.So, 480 + 19.8 = 499.8.Therefore, total is 30,000 + 499.8 = 30,499.8.So, sqrt(930,240,000) ≈ 30,499.8.But when I compute (30,499.8)², I get 929,238,000.04, which is way less than 930,240,000. That can't be right.Wait, no, that must be a miscalculation because 30,499.8 is approximately the square root, so (30,499.8)² should be approximately 930,240,000.Wait, let me compute 30,499.8 × 30,499.8.Let me write it as (30,500 - 0.2) × (30,500 - 0.2) = 30,500² - 2×30,500×0.2 + (0.2)².Compute each term:30,500² = 930,250,000.2×30,500×0.2 = 2×30,500×0.2 = 12,200.(0.2)² = 0.04.So, (30,500 - 0.2)² = 930,250,000 - 12,200 + 0.04 = 930,250,000 - 12,200 = 929,238,000 + 0.04 = 929,238,000.04.Wait, that's correct, but that's way less than 930,240,000.So, that means my approximation is wrong. There's a mistake in my reasoning.Wait, no, actually, sqrt(930,240,000) is 30,499.8, but when I square 30,499.8, I get 929,238,000.04, which is way less than 930,240,000. That can't be.Wait, that must mean that my earlier factorization was wrong.Wait, let me go back.I had 930,240,000 = 16 × 5,814 × 10,000.Wait, 16 × 5,814 is 93,024, and 93,024 × 10,000 is 930,240,000.Then, sqrt(930,240,000) = sqrt(93,024 × 10,000) = sqrt(93,024) × 100.Then, sqrt(93,024) = sqrt(16 × 5,814) = 4 × sqrt(5,814).Then, sqrt(5,814) = sqrt(2 × 3² × 17 × 19) = 3 × sqrt(2 × 17 × 19) = 3 × sqrt(646).So, sqrt(93,024) = 4 × 3 × sqrt(646) = 12 × sqrt(646).Therefore, sqrt(930,240,000) = 12 × sqrt(646) × 100 = 1,200 × sqrt(646).So, sqrt(646) ≈ 25.4165, so 1,200 × 25.4165 ≈ 30,499.8.But when I compute (30,499.8)², I get 929,238,000.04, which is way less than 930,240,000.Wait, that can't be. There must be a miscalculation in the factorization.Wait, let me compute sqrt(930,240,000) directly.Compute sqrt(930,240,000).Let me note that 930,240,000 = 930,240 × 1,000.So, sqrt(930,240,000) = sqrt(930,240 × 1,000) = sqrt(930,240) × sqrt(1,000).Compute sqrt(930,240) and sqrt(1,000).sqrt(1,000) ≈ 31.6227766.Now, compute sqrt(930,240).Let me factor 930,240.930,240 ÷ 10 = 93,024.So, 930,240 = 93,024 × 10.We already factored 93,024 earlier as 16 × 5,814, which is 16 × 2 × 3² × 17 × 19.So, 930,240 = 16 × 2 × 3² × 17 × 19 × 10.Wait, 10 is 2 × 5, so overall, 930,240 = 2⁵ × 3² × 5 × 17 × 19.Therefore, sqrt(930,240) = sqrt(2⁵ × 3² × 5 × 17 × 19) = 2² × 3 × sqrt(2 × 5 × 17 × 19) = 4 × 3 × sqrt(3,230) = 12 × sqrt(3,230).Wait, sqrt(3,230). Hmm, 3,230 is between 56²=3,136 and 57²=3,249. So, sqrt(3,230) is between 56 and 57.Compute 56.8²: 56²=3,136, 0.8²=0.64, 2×56×0.8=89.6. So, 56.8²=3,136 + 89.6 + 0.64=3,226.24.That's close to 3,230. So, 56.8²=3,226.24.Difference is 3,230 - 3,226.24=3.76.So, approximate sqrt(3,230)=56.8 + 3.76/(2×56.8)=56.8 + 3.76/113.6≈56.8 + 0.033≈56.833.Therefore, sqrt(3,230)≈56.833.Therefore, sqrt(930,240)=12×56.833≈12×56.833≈682.Wait, 12×56=672, 12×0.833≈10, so total≈682.Therefore, sqrt(930,240)≈682.Therefore, sqrt(930,240,000)=sqrt(930,240)×sqrt(1,000)≈682×31.6227766≈682×31.6227766.Compute 682×30=20,460.682×1.6227766≈682×1.6=1,091.2, and 682×0.0227766≈15.5.So, total≈20,460 + 1,091.2 + 15.5≈20,460 + 1,106.7≈21,566.7.Wait, that can't be right because 21,566.7² is way more than 930,240,000.Wait, no, wait, sqrt(930,240,000) is approximately 30,499.8, as we had earlier, but this method is giving me 21,566.7, which is way off. So, I must have made a mistake in this approach.Wait, no, actually, sqrt(930,240,000) is sqrt(930,240 × 1,000) = sqrt(930,240) × sqrt(1,000). But sqrt(930,240) is approximately 964.468, because 964²=929,296 and 965²=931,225. So, sqrt(930,240) is between 964 and 965.Compute 964.5²: 964²=929,296, 0.5²=0.25, 2×964×0.5=964. So, 964.5²=929,296 + 964 + 0.25=930,260.25.That's very close to 930,240. So, sqrt(930,240)= approximately 964.5 - (930,260.25 - 930,240)/(2×964.5).Compute difference: 930,260.25 - 930,240=20.25.So, sqrt(930,240)≈964.5 - 20.25/(2×964.5)=964.5 - 20.25/1929≈964.5 - 0.0105≈964.4895.Therefore, sqrt(930,240)≈964.4895.Therefore, sqrt(930,240,000)=sqrt(930,240)×sqrt(1,000)=964.4895×31.6227766≈964.4895×31.6227766.Compute 964×30=28,920.964×1.6227766≈964×1.6=1,542.4, and 964×0.0227766≈21.88.So, total≈28,920 + 1,542.4 + 21.88≈28,920 + 1,564.28≈30,484.28.Then, add the 0.4895×31.6227766≈15.44.So, total≈30,484.28 + 15.44≈30,499.72.Therefore, sqrt(930,240,000)≈30,499.72, which is approximately 30,499.72.So, that's consistent with our earlier approximation of 30,499.8.Therefore, sqrt(930,240,000)≈30,499.72, so the integer part is 30,499.Therefore, the maximum integer n is 30,499.But just to make sure, let me compute 30,499² and 30,500² again.30,499²: As computed earlier, it's 930,189,001.30,500²: 930,250,000.Our target is 930,240,000.So, 30,499² = 930,189,001 < 930,240,000 < 30,500² = 930,250,000.Therefore, 30,499 is the largest integer n such that n² ≤ 930,240,000.Therefore, the answer is 30,499.**Final Answer**The maximum positive integer ( n ) is boxed{30499}.