answer：I need to factorize the expression (3a^{2} - 27).First, I notice that both terms have a common factor of 3. I can factor out the 3:[3a^{2} - 27 = 3(a^{2} - 9)]Next, I recognize that (a^{2} - 9) is a difference of squares, since (a^{2}) is a square of (a) and 9 is a square of 3. The difference of squares can be factored as:[a^{2} - 9 = (a + 3)(a - 3)]Substituting this back into the expression, I get:[3(a^{2} - 9) = 3(a + 3)(a - 3)]So, the factorized form of (3a^{2} - 27) is (3(a + 3)(a - 3)).

answer：Okay, so I have this geometry problem here about an isosceles triangle ABC. Let me try to visualize it first. The triangle has sides AB and BC each of length 125, and the base AC is 150 units long. Point D is the midpoint of AC, so that should be straightforward. Then, points E and F are on sides BC and AB respectively, such that DE is perpendicular to BC and DF is perpendicular to AB. I need to find the area of triangle DEF.Hmm, okay. Since it's an isosceles triangle with AB = BC = 125 and AC = 150, I think it would help to assign coordinates to the points to make things clearer. Let me place point A at (0, 0) and point C at (150, 0). That way, AC lies along the x-axis, which should simplify calculations.Now, point B is somewhere above the x-axis. Since AB = BC = 125, point B must be equidistant from A and C. That means it should lie somewhere along the perpendicular bisector of AC. The midpoint of AC is at (75, 0), so the perpendicular bisector would be the line x = 75. So, point B has coordinates (75, y) for some y.To find y, I can use the distance formula. The distance from A to B is 125, so:√[(75 - 0)^2 + (y - 0)^2] = 125 √[5625 + y^2] = 125 Squaring both sides: 5625 + y^2 = 15625 So, y^2 = 15625 - 5625 = 10000 Therefore, y = 100 (since it's above the x-axis)So, point B is at (75, 100).Now, point D is the midpoint of AC, which is at (75, 0).Next, I need to find points E and F. Let's start with E on BC such that DE is perpendicular to BC.First, let's find the equation of line BC. Points B and C are at (75, 100) and (150, 0) respectively. The slope of BC is:m_BC = (0 - 100)/(150 - 75) = (-100)/75 = -4/3So, the equation of BC is:y - 100 = (-4/3)(x - 75)Simplifying:y = (-4/3)x + 100 + (4/3)*75 y = (-4/3)x + 100 + 100 y = (-4/3)x + 200Now, DE is perpendicular to BC, so the slope of DE is the negative reciprocal of -4/3, which is 3/4.Since D is at (75, 0), the equation of DE is:y - 0 = (3/4)(x - 75) y = (3/4)x - (225/4)Now, point E lies on both BC and DE, so let's set the equations equal to each other:(3/4)x - (225/4) = (-4/3)x + 200To solve for x, let's eliminate the fractions by multiplying both sides by 12:12*(3/4)x - 12*(225/4) = 12*(-4/3)x + 12*200 9x - 675 = -16x + 2400Bring all terms to one side:9x + 16x = 2400 + 675 25x = 3075 x = 3075 / 25 x = 123Now, plug x = 123 into the equation of DE to find y:y = (3/4)*123 - (225/4) y = (369/4) - (225/4) y = (144)/4 y = 36Wait, that doesn't seem right because when I plug x = 123 into BC's equation:y = (-4/3)*123 + 200 y = -164 + 200 y = 36Okay, so point E is at (123, 36). Hmm, but earlier I thought DE was perpendicular to BC, but when I calculated E, it's not on the x-axis. Wait, maybe I made a mistake in my calculation.Wait, no, point E is on BC, which is from (75, 100) to (150, 0). So, E is somewhere along that line, not necessarily on the x-axis. So, (123, 36) is correct.Wait, but earlier, when I calculated DE, I thought it was from D(75, 0) to E(123, 36). Let me check the slope:Slope of DE: (36 - 0)/(123 - 75) = 36/48 = 3/4, which is correct because it's perpendicular to BC's slope of -4/3.Okay, so E is at (123, 36). Got it.Now, let's find point F on AB such that DF is perpendicular to AB.First, let's find the equation of AB. Points A(0, 0) and B(75, 100). The slope of AB is:m_AB = (100 - 0)/(75 - 0) = 100/75 = 4/3So, the equation of AB is:y = (4/3)xSince DF is perpendicular to AB, its slope is the negative reciprocal, which is -3/4.Point D is at (75, 0), so the equation of DF is:y - 0 = (-3/4)(x - 75) y = (-3/4)x + (225/4)Now, point F lies on both AB and DF, so set their equations equal:(4/3)x = (-3/4)x + (225/4)Multiply both sides by 12 to eliminate fractions:16x = -9x + 675 16x + 9x = 675 25x = 675 x = 27Now, plug x = 27 into AB's equation to find y:y = (4/3)*27 = 36So, point F is at (27, 36).Wait, that's interesting. Both E and F have the same y-coordinate, 36. So, points E(123, 36) and F(27, 36) are both at y = 36.Now, I need to find the area of triangle DEF, where D is at (75, 0), E is at (123, 36), and F is at (27, 36).Let me plot these points mentally. D is at (75, 0), E is at (123, 36), and F is at (27, 36). So, triangle DEF has vertices at (75, 0), (123, 36), and (27, 36).Hmm, since both E and F are at y = 36, the line EF is horizontal. The base of the triangle DEF can be considered as EF, and the height would be the vertical distance from D to this line, which is 36 units.But wait, actually, since D is at (75, 0), the vertical distance from D to EF is 36 units. So, the area would be (base * height)/2.First, let's find the length of EF. Since E is at (123, 36) and F is at (27, 36), the distance between them is:EF = |123 - 27| = 96 units.So, the base is 96, and the height is 36. Therefore, the area is:Area = (96 * 36)/2 = (3456)/2 = 1728.Wait, that seems too large. Let me double-check.Alternatively, I can use the coordinates to calculate the area using the shoelace formula.Shoelace formula for area of triangle with vertices (x1,y1), (x2,y2), (x3,y3):Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Plugging in the points D(75, 0), E(123, 36), F(27, 36):Area = |75*(36 - 36) + 123*(36 - 0) + 27*(0 - 36)| / 2 = |0 + 123*36 + 27*(-36)| / 2 = |0 + 4428 - 972| / 2 = |3456| / 2 = 1728Hmm, same result. But I feel like this might be incorrect because the triangle seems too large. Let me think again.Wait, maybe I made a mistake in assigning coordinates. Let me re-examine.Point E is on BC, which goes from (75, 100) to (150, 0). I found E at (123, 36). Let me confirm that.Equation of BC: y = (-4/3)x + 200At x = 123, y = (-4/3)*123 + 200 = -164 + 200 = 36. Correct.Equation of DE: y = (3/4)x - 225/4At x = 123, y = (3/4)*123 - 225/4 = 92.25 - 56.25 = 36. Correct.Similarly, for point F on AB: equation of AB is y = (4/3)xEquation of DF: y = (-3/4)x + 225/4Setting equal: (4/3)x = (-3/4)x + 225/4Multiply both sides by 12: 16x = -9x + 675 => 25x = 675 => x = 27, y = 36. Correct.So, points E and F are correctly found.Now, triangle DEF has vertices at (75,0), (123,36), (27,36). So, it's a triangle with a base EF of 96 units and height from D to EF of 36 units. So, area is (96*36)/2 = 1728.But wait, the original triangle ABC has sides 125, 125, 150. Let me calculate its area to see if 1728 is plausible.Area of ABC: Since it's isosceles, height can be found by splitting the base AC into two 75 units each.Height h from B to AC: Using Pythagoras in triangle ABD (where D is midpoint):h^2 + 75^2 = 125^2 h^2 = 15625 - 5625 = 10000 h = 100So, area of ABC is (150 * 100)/2 = 7500.So, area of DEF is 1728, which is about 23% of ABC's area. That seems plausible.Wait, but the problem says "Find the area of triangle DEF." So, maybe 1728 is the correct answer. But let me check my calculations again.Alternatively, maybe I can use vectors or coordinate geometry differently.Another approach: Since D is the midpoint of AC, which is at (75,0). E is on BC at (123,36), F is on AB at (27,36).So, vectors DE and DF can be found, and then the area is half the magnitude of the cross product.Vector DE = E - D = (123 - 75, 36 - 0) = (48, 36)Vector DF = F - D = (27 - 75, 36 - 0) = (-48, 36)Cross product DE x DF = (48)(36) - (36)(-48) = 1728 + 1728 = 3456Area = |3456| / 2 = 1728Same result. So, it seems consistent.Wait, but I'm getting 1728, but the problem might expect a different answer. Let me check if I made a mistake in interpreting the problem.Wait, the problem says "E is on side BC so that BC and DE are perpendicular." Similarly for F.Wait, in my calculation, DE is perpendicular to BC, which is correct because the slope of DE is 3/4 and BC is -4/3, which are negative reciprocals. Similarly, DF is perpendicular to AB, which has slope 4/3, and DF has slope -3/4, which is correct.So, everything seems correct. Therefore, the area of triangle DEF is 1728.Wait, but 1728 is a large number. Let me think again about the coordinates.Point E is at (123,36), which is 123 units along the x-axis from A, which is at 0. Since AC is 150 units, 123 is within the range. Similarly, F is at (27,36), which is within AB.Wait, but when I calculated the area using the shoelace formula, I got 1728, which is correct based on the coordinates. So, maybe that's the answer.Alternatively, maybe I can use coordinate geometry differently. Let me try to find the lengths of DE and DF and then use Heron's formula.Length DE: distance between D(75,0) and E(123,36):DE = sqrt[(123-75)^2 + (36-0)^2] = sqrt[48^2 + 36^2] = sqrt[2304 + 1296] = sqrt[3600] = 60Similarly, length DF: distance between D(75,0) and F(27,36):DF = sqrt[(27-75)^2 + (36-0)^2] = sqrt[(-48)^2 + 36^2] = sqrt[2304 + 1296] = sqrt[3600] = 60So, DE = DF = 60 units.Now, the distance between E and F is 96 units, as calculated earlier.So, triangle DEF has sides DE = 60, DF = 60, EF = 96.Wait, so it's an isosceles triangle with two sides of 60 and base 96.We can calculate its area using Heron's formula.Perimeter semi-perimeter, s = (60 + 60 + 96)/2 = 216/2 = 108Area = sqrt[s(s - a)(s - b)(s - c)] = sqrt[108*(108 - 60)*(108 - 60)*(108 - 96)] = sqrt[108*48*48*12]Calculate inside the sqrt:108*12 = 1296 48*48 = 2304 So, 1296 * 2304Wait, that's a big number. Let me compute step by step:First, 108*48 = 5184 Then, 5184*48 = 248832 Then, 248832*12 = 2985984So, sqrt[2985984] = 1728So, area is 1728. Same result.Therefore, despite the large number, it seems correct.Wait, but let me think again. The original triangle ABC has area 7500, and DEF is 1728, which is about 23% of ABC. That seems reasonable.Alternatively, maybe I can use vectors or parametric equations, but I think the coordinate approach is solid.So, after all these calculations, I think the area of triangle DEF is 1728.

answer：Okay, so I need to find the sum of all the positive factors of 120. Hmm, where do I start? I remember that to find the sum of factors, there's something to do with prime factorization. Let me think about that.First, I should probably break down 120 into its prime factors. I know that 120 is an even number, so it's divisible by 2. Let's divide 120 by 2: 120 ÷ 2 = 60. Okay, 60 is still even, so divide by 2 again: 60 ÷ 2 = 30. Still even, so divide by 2 once more: 30 ÷ 2 = 15. Now, 15 isn't even, so I'll try the next prime number, which is 3. Dividing 15 by 3 gives me 5. And 5 is a prime number itself, so I'm done with the prime factorization.So, putting it all together, 120 can be expressed as 2 × 2 × 2 × 3 × 5, which is the same as 2³ × 3¹ × 5¹. Got that part down.Now, I recall there's a formula for finding the sum of all positive divisors of a number based on its prime factorization. If a number N can be written as N = p₁^a × p₂^b × p₃^c × ... where p₁, p₂, p₃, etc., are prime numbers and a, b, c, etc., are their respective exponents, then the sum of the divisors of N is given by:(1 + p₁ + p₁² + ... + p₁^a) × (1 + p₂ + p₂² + ... + p₂^b) × (1 + p₃ + p₃² + ... + p₃^c) × ...So, applying this to 120, which is 2³ × 3¹ × 5¹, the sum of its divisors should be:(1 + 2 + 2² + 2³) × (1 + 3) × (1 + 5)Let me calculate each part step by step.First, for the prime number 2 with exponent 3:1 + 2 + 4 + 8 = 15Okay, that's the sum for the powers of 2.Next, for the prime number 3 with exponent 1:1 + 3 = 4That's straightforward.Lastly, for the prime number 5 with exponent 1:1 + 5 = 6Alright, so now I have three numbers: 15, 4, and 6. I need to multiply them together to get the total sum of the divisors.Let's do that:15 × 4 = 60Then, 60 × 6 = 360So, the sum of all the positive factors of 120 is 360.Wait, let me double-check to make sure I didn't make any mistakes. Maybe I should list out all the factors of 120 and add them up to confirm.Starting with 1, since 1 is a factor of every number.Then, 2, because 120 is even.3, since the sum of the digits (1+2+0=3) is divisible by 3.4, because 120 ÷ 4 = 30, which is an integer.5, since it ends with a 0.6, because it's divisible by both 2 and 3.8, since 120 ÷ 8 = 15.10, because it ends with a 0.12, since 120 ÷ 12 = 10.15, as we saw earlier.20, because 120 ÷ 20 = 6.24, since 120 ÷ 24 = 5.30, because 120 ÷ 30 = 4.40, as 120 ÷ 40 = 3.60, since 120 ÷ 60 = 2.And finally, 120 itself.So, listing all the factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.Now, let's add them up:1 + 2 = 33 + 3 = 66 + 4 = 1010 + 5 = 1515 + 6 = 2121 + 8 = 2929 + 10 = 3939 + 12 = 5151 + 15 = 6666 + 20 = 8686 + 24 = 110110 + 30 = 140140 + 40 = 180180 + 60 = 240240 + 120 = 360Okay, so adding them all up also gives me 360. That matches the result I got using the formula. So, it looks like my initial calculation was correct.I think I understand the process now. By breaking down the number into its prime factors and applying the sum of divisors formula, I can efficiently find the sum without having to list out all the factors, which could be time-consuming for larger numbers.Just to recap, the steps are:1. Prime factorize the number.2. Use the sum of divisors formula based on the exponents of the prime factors.3. Multiply the sums together to get the total.This method seems reliable and efficient, especially for larger numbers where listing all factors would be impractical.I wonder if there's a way to remember this formula better. Maybe by associating it with the prime factorization process, since they go hand in hand. Also, practicing with a few more examples might help solidify my understanding.Another thing I'm thinking about is why this formula works. It must have something to do with the multiplicative property of divisors. Each divisor can be represented as a combination of the prime factors raised to different powers, and by summing all these combinations, we get the total sum of the divisors.For example, in the case of 120, each divisor can be written as 2^a × 3^b × 5^c, where a can be 0, 1, 2, or 3; b can be 0 or 1; and c can be 0 or 1. So, when we take the product of (1 + 2 + 4 + 8), (1 + 3), and (1 + 5), we're essentially considering all possible combinations of these exponents, which correspond to all the divisors of 120.This makes sense because each term in the product represents the contribution of each prime factor to the overall sum. By multiplying these together, we account for every possible combination of exponents, thereby covering all divisors.It's interesting how mathematical formulas often have such elegant justifications. Understanding why a formula works can make it easier to remember and apply correctly.I should also consider potential mistakes that could happen when applying this formula. For instance, forgetting to include 1 in the sum for each prime factor could lead to an incorrect result. Also, miscounting the exponents during prime factorization would throw off the entire calculation.To avoid these errors, it's important to carefully perform each step. First, ensure that the prime factorization is correct by double-checking the division process. Then, when applying the formula, make sure to include all terms from 1 up to the highest power of each prime factor.Another point to consider is that this formula works for any positive integer, not just composite numbers. For prime numbers, the sum of divisors would simply be 1 plus the prime itself, since primes have no other divisors.Thinking about edge cases can help in understanding the formula better. For example, let's take the number 1. Its prime factorization is just 1, but since 1 isn't a prime number, the formula would technically be (1) for each prime factor, which in this case, there are none. So, the sum of divisors of 1 is just 1, which makes sense.Similarly, for a prime number like 7, the sum of divisors would be (1 + 7) = 8, which is correct because the only divisors are 1 and 7.This consistency across different types of numbers adds to the robustness of the formula.I also recall that there's a related concept called the "aliquot sum," which is the sum of all proper divisors of a number, excluding the number itself. For 120, the aliquot sum would be 360 - 120 = 240. Knowing this, I can see how the sum of divisors formula is a fundamental tool in number theory, with applications in various areas like perfect numbers, abundant numbers, and deficient numbers.A perfect number is one where the sum of its proper divisors equals the number itself. For example, 6 is a perfect number because its proper divisors are 1, 2, and 3, which add up to 6. An abundant number has a sum of proper divisors greater than the number, like 12, whose proper divisors sum to 16. A deficient number has a sum of proper divisors less than the number, such as 8, whose proper divisors sum to 7.Understanding these classifications can be useful in deeper studies of number theory and can even have applications in fields like cryptography and algorithm design.Going back to the original problem, I'm confident that the sum of the positive factors of 120 is indeed 360. The two methods I used—the formula based on prime factorization and manually listing and adding the factors—both led me to the same result. This consistency gives me assurance that my answer is correct.In summary, the process involves:1. Prime factorization of the number.2. Applying the sum of divisors formula.3. Verifying the result by alternative methods if necessary.This structured approach not only solves the problem at hand but also reinforces my understanding of number theory concepts, making it easier to tackle similar problems in the future.

answer：Okay, so I have this problem where a right circular cylinder is inscribed in a right circular cone. The cone has a diameter of 12 and an altitude of 18. The axes of both the cylinder and the cone coincide, which means they share the same central axis. Additionally, it's given that the diameter of the cylinder is equal to its height. I need to find the radius of the cylinder and express it as a common fraction.Alright, let me start by visualizing the problem. I imagine a cone standing upright with its pointy end at the bottom and the base on top. Inside this cone, there's a cylinder touching the sides of the cone. The cylinder's top and bottom are parallel to the base of the cone, and their centers lie along the central axis of the cone.First, let's note down the given information:- Diameter of the cone: 12, so the radius of the cone's base is half of that, which is 6.- Altitude (height) of the cone: 18 units.- The cylinder is inscribed in the cone, meaning it touches the cone exactly at some points along its curved surface.- The diameter of the cylinder is equal to its height. Since the diameter is twice the radius, if I let the radius of the cylinder be ( r ), then its diameter is ( 2r ), and its height is also ( 2r ).So, my goal is to find ( r ).I remember that when dealing with similar shapes, especially cones and inscribed cylinders, similar triangles can be really helpful. Let me try to set up a coordinate system or a diagram to model this.Let's consider a vertical cross-section through the axis of the cone and the cylinder. This cross-section will be a triangle representing the cone and a smaller, similar triangle representing the space above the cylinder inside the cone.The large triangle (the cone) has a height of 18 and a base radius of 6. The smaller triangle (the part of the cone above the cylinder) will have a height of ( 18 - 2r ) because the cylinder's height is ( 2r ), and the radius of this smaller triangle will be equal to the radius of the cylinder, which is ( r ).Since these two triangles are similar (they have the same shape but different sizes), their corresponding dimensions will be proportional. That is, the ratio of their heights will be equal to the ratio of their bases.Mathematically, this can be written as:[frac{text{Height of smaller triangle}}{text{Height of larger triangle}} = frac{text{Base of smaller triangle}}{text{Base of larger triangle}}]Plugging in the known values:[frac{18 - 2r}{18} = frac{r}{6}]Wait, hold on. Let me make sure I got that right. The height of the smaller triangle is ( 18 - 2r ), and its base is ( r ). The height of the larger triangle is 18, and its base is 6. So yes, the proportion should be:[frac{18 - 2r}{18} = frac{r}{6}]Hmm, but I think I might have mixed up the terms. Let me think again. The smaller triangle is similar to the larger triangle, so the ratio of their corresponding sides should be equal. The height of the smaller triangle is ( 18 - 2r ), and the height of the larger triangle is 18. The base of the smaller triangle is ( r ), and the base of the larger triangle is 6.So, the ratio of the heights is ( frac{18 - 2r}{18} ), and the ratio of the bases is ( frac{r}{6} ). Since the triangles are similar, these ratios should be equal:[frac{18 - 2r}{18} = frac{r}{6}]Yes, that seems correct.Now, let's solve for ( r ). First, cross-multiply to eliminate the fractions:[6(18 - 2r) = 18r]Let me compute the left side:[6 times 18 = 108][6 times (-2r) = -12r]So, the left side becomes ( 108 - 12r ).Now, the equation is:[108 - 12r = 18r]Let me bring all terms to one side. I'll add ( 12r ) to both sides:[108 = 18r + 12r][108 = 30r]Now, divide both sides by 30:[r = frac{108}{30}]Simplify the fraction by dividing numerator and denominator by 6:[r = frac{18}{5}]So, the radius of the cylinder is ( frac{18}{5} ).Wait, let me verify this result. If ( r = frac{18}{5} ), then the diameter of the cylinder is ( 2r = frac{36}{5} ), and the height is also ( frac{36}{5} ). The height of the smaller triangle above the cylinder would be ( 18 - frac{36}{5} = frac{90}{5} - frac{36}{5} = frac{54}{5} ).Now, let's check the ratio of the heights and the ratio of the bases. The ratio of the heights is ( frac{frac{54}{5}}{18} = frac{54}{5 times 18} = frac{54}{90} = frac{3}{5} ). The ratio of the bases is ( frac{frac{18}{5}}{6} = frac{18}{5 times 6} = frac{18}{30} = frac{3}{5} ). Since both ratios are equal, the result is consistent.Therefore, the radius of the cylinder is indeed ( frac{18}{5} ).**Final Answer**The radius of the cylinder is boxed{dfrac{18}{5}}.

answer：Okay, so I have this problem about a parabola described by the equation ( y = ax^2 + bx + c ). The vertex is at (2, 1), and it passes through the point (0, 5). I need to find ( a + b - c ). Hmm, let me think about how to approach this step by step.First, I remember that the vertex form of a parabola is useful when we know the vertex. The vertex form is ( y = a(x - h)^2 + k ), where (h, k) is the vertex. In this case, the vertex is (2, 1), so substituting those values in, the equation becomes ( y = a(x - 2)^2 + 1 ). That seems right.Now, I need to find the value of 'a'. To do that, I can use the fact that the parabola passes through the point (0, 5). That means when x is 0, y is 5. So I can substitute x = 0 and y = 5 into the equation and solve for 'a'.Let me write that out:( 5 = a(0 - 2)^2 + 1 )Simplifying the expression inside the parentheses first:( 0 - 2 = -2 )So, squaring that:( (-2)^2 = 4 )Now, plug that back into the equation:( 5 = 4a + 1 )Hmm, okay, so now I can solve for 'a'. Subtract 1 from both sides:( 5 - 1 = 4a )Which simplifies to:( 4 = 4a )Divide both sides by 4:( a = 1 )Alright, so 'a' is 1. Now, I can write the equation of the parabola in vertex form:( y = 1(x - 2)^2 + 1 )But the problem gives the equation in standard form ( y = ax^2 + bx + c ), so I need to expand this vertex form to get it into standard form.Let me expand ( (x - 2)^2 ):( (x - 2)^2 = x^2 - 4x + 4 )So, substituting back into the equation:( y = 1(x^2 - 4x + 4) + 1 )Distribute the 1:( y = x^2 - 4x + 4 + 1 )Combine like terms:( y = x^2 - 4x + 5 )So, now the equation is in standard form, and I can identify the coefficients:- ( a = 1 )- ( b = -4 )- ( c = 5 )The question asks for ( a + b - c ). Let's compute that:( a + b - c = 1 + (-4) - 5 )Simplify:( 1 - 4 = -3 )Then,( -3 - 5 = -8 )So, ( a + b - c = -8 ).Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the vertex form, plugging in the point (0,5) gave me 'a' as 1, which seems correct. Expanding the vertex form to standard form also looks right: ( x^2 - 4x + 5 ). The coefficients are indeed 1, -4, and 5. Adding and subtracting them as per the question gives -8. I think that's correct.Just to be thorough, maybe I can verify by plugging in another point. The vertex is at (2,1), so let's see if that satisfies the equation ( y = x^2 - 4x + 5 ).Substitute x = 2:( y = (2)^2 - 4*(2) + 5 = 4 - 8 + 5 = 1 )Yes, that gives y = 1, which matches the vertex. Also, plugging in x = 0 gives y = 5, which is the point given. So, the equation seems correct.Therefore, I'm confident that ( a + b - c = -8 ).**Final Answer**The value of ( a + b - c ) is boxed{-8}.

answer：Alright, so I have this problem where the general term of a sequence is given by ( a_n = n^3 - n ) where ( n ) is a positive integer. I need to find the sequence formed by arranging the terms of ( {a_n} ) that end in the digit 0 in increasing order, which is denoted as ( {b_n} ). Then, I have to determine the remainder when ( b_{2018} ) is divided by 7.First, let me understand what ( a_n = n^3 - n ) represents. I can factor this as ( a_n = n(n^2 - 1) = n(n - 1)(n + 1) ). So, it's the product of three consecutive integers: ( n - 1 ), ( n ), and ( n + 1 ). Since we're interested in terms that end with the digit 0, we need ( a_n ) to be divisible by 10. A number is divisible by 10 if it has both 2 and 5 as factors. So, I need to find values of ( n ) such that ( n(n - 1)(n + 1) ) is divisible by both 2 and 5.Looking at the product ( n(n - 1)(n + 1) ), it's the product of three consecutive integers, so it's always divisible by 2 and 3. But we need it to be divisible by 5 as well. So, one of the three consecutive numbers must be divisible by 5. Therefore, ( n ), ( n - 1 ), or ( n + 1 ) must be divisible by 5. That means ( n ) must be congruent to 0, 1, or 4 modulo 5. So, ( n equiv 0, 1, 4 mod 5 ).But we also need the term to end with 0, which means it must be divisible by 10. Since the product is already divisible by 2, we just need to ensure it's divisible by 5, which we've already covered. So, the values of ( n ) that make ( a_n ) end with 0 are those where ( n equiv 0, 1, 4 mod 5 ).Wait, but let me check that. If ( n equiv 0 mod 5 ), then ( n ) is divisible by 5. If ( n equiv 1 mod 5 ), then ( n - 1 ) is divisible by 5. If ( n equiv 4 mod 5 ), then ( n + 1 ) is divisible by 5. So, yes, those are the cases where ( a_n ) is divisible by 5, and since it's also divisible by 2, it's divisible by 10, meaning it ends with 0.But wait, actually, in the problem statement, it's not just about divisibility by 10, but specifically the unit digit being 0. So, I think my reasoning is correct because if a number is divisible by 10, its unit digit is 0.Now, I need to find the sequence ( {b_n} ) which is formed by arranging the terms of ( {a_n} ) that end in 0 in increasing order. So, I need to list all ( a_n ) where ( a_n ) ends with 0, and then sort them in increasing order.But since ( a_n ) is a cubic function, it's going to grow quite rapidly. So, the terms ( b_n ) are going to be the values of ( a_n ) for ( n ) such that ( n equiv 0, 1, 4 mod 5 ), sorted in increasing order.But wait, actually, let me think again. The problem says "the sequence formed by arranging the terms of ( {a_n} ) that end in the digit 0 in increasing order." So, it's not necessarily that ( n ) is arranged in a particular way, but rather the terms ( a_n ) that end with 0 are arranged in increasing order.So, I need to find all ( a_n ) such that ( a_n ) ends with 0, collect them, sort them, and then find the 2018th term in this sorted sequence, and then find its remainder when divided by 7.But this seems complicated because ( a_n ) is a cubic function, so the terms grow quickly, and it's not straightforward to list them all. So, perhaps there's a pattern or periodicity modulo 7 that I can exploit.Wait, the question is about the remainder when ( b_{2018} ) is divided by 7. So, maybe instead of computing ( b_{2018} ) directly, I can find a pattern in the remainders of ( a_n ) modulo 7 for the values of ( n ) that make ( a_n ) end with 0.So, perhaps I can find the values of ( n ) such that ( a_n ) ends with 0, and then find the corresponding ( a_n mod 7 ), and see if there's a repeating cycle or pattern.Alternatively, maybe I can find that the sequence ( {b_n} ) modulo 7 is periodic, and then find the period, and then compute ( 2018 mod text{period} ) to find the equivalent smaller index.But let's try to approach this step by step.First, let's find the values of ( n ) such that ( a_n ) ends with 0. As I reasoned earlier, ( n equiv 0, 1, 4 mod 5 ). So, every 5 numbers, there are 3 such ( n ) that make ( a_n ) end with 0.But wait, actually, let's check for ( n ) from 1 to 10:- ( n = 1 ): ( a_1 = 1 - 1 = 0 ) → ends with 0- ( n = 2 ): ( 8 - 2 = 6 ) → doesn't end with 0- ( n = 3 ): ( 27 - 3 = 24 ) → doesn't end with 0- ( n = 4 ): ( 64 - 4 = 60 ) → ends with 0- ( n = 5 ): ( 125 - 5 = 120 ) → ends with 0- ( n = 6 ): ( 216 - 6 = 210 ) → ends with 0- ( n = 7 ): ( 343 - 7 = 336 ) → doesn't end with 0- ( n = 8 ): ( 512 - 8 = 504 ) → doesn't end with 0- ( n = 9 ): ( 729 - 9 = 720 ) → ends with 0- ( n = 10 ): ( 1000 - 10 = 990 ) → ends with 0So, in the first 10 terms, ( n = 1, 4, 5, 6, 9, 10 ) produce ( a_n ) ending with 0. That's 6 terms, not 3. So, my earlier reasoning was incorrect. It's not 3 terms every 5 numbers, but rather 6 terms every 10 numbers.Wait, so in the first 10 numbers, 6 values of ( n ) produce ( a_n ) ending with 0. So, perhaps the pattern repeats every 10 numbers, with 6 such terms each time.Looking at the next set, ( n = 11 ) to ( n = 20 ):- ( n = 11 ): ( 1331 - 11 = 1320 ) → ends with 0- ( n = 12 ): ( 1728 - 12 = 1716 ) → doesn't end with 0- ( n = 13 ): ( 2197 - 13 = 2184 ) → doesn't end with 0- ( n = 14 ): ( 2744 - 14 = 2730 ) → ends with 0- ( n = 15 ): ( 3375 - 15 = 3360 ) → ends with 0- ( n = 16 ): ( 4096 - 16 = 4080 ) → ends with 0- ( n = 17 ): ( 4913 - 17 = 4896 ) → doesn't end with 0- ( n = 18 ): ( 5832 - 18 = 5814 ) → doesn't end with 0- ( n = 19 ): ( 6859 - 19 = 6840 ) → ends with 0- ( n = 20 ): ( 8000 - 20 = 7980 ) → ends with 0Again, ( n = 11, 14, 15, 16, 19, 20 ) produce ( a_n ) ending with 0. So, 6 terms every 10 numbers.So, the pattern is that in every block of 10 consecutive integers, there are 6 values of ( n ) such that ( a_n ) ends with 0. Therefore, the sequence ( {b_n} ) is formed by taking these 6 terms from each block of 10, and arranging them in increasing order.But wait, actually, the terms ( a_n ) are not necessarily increasing within each block. For example, ( a_1 = 0 ), ( a_4 = 60 ), ( a_5 = 120 ), ( a_6 = 210 ), ( a_9 = 720 ), ( a_{10} = 990 ). So, within the first block, the terms are increasing as ( n ) increases.Similarly, in the next block, ( a_{11} = 1320 ), ( a_{14} = 2730 ), ( a_{15} = 3360 ), ( a_{16} = 4080 ), ( a_{19} = 6840 ), ( a_{20} = 7980 ). Again, increasing as ( n ) increases.So, in each block of 10, the 6 terms are increasing as ( n ) increases. Therefore, the overall sequence ( {b_n} ) is just the concatenation of these blocks, each contributing 6 terms in increasing order.Therefore, the sequence ( {b_n} ) is ordered such that each block of 10 ( n ) values contributes 6 terms, and within each block, the terms are in increasing order. Therefore, the entire sequence ( {b_n} ) is simply the terms ( a_n ) for ( n ) congruent to 1,4,5,6,9,10 modulo 10, in increasing order of ( n ).Wait, but actually, when arranging the terms in increasing order, it's not just the order of ( n ), but the actual values of ( a_n ). However, since ( a_n ) is increasing for ( n geq 1 ), the terms ( a_n ) for ( n ) in each block will be larger than the terms in the previous block. So, the sequence ( {b_n} ) is just the concatenation of these blocks, each contributing 6 terms in increasing order.Therefore, the first 6 terms of ( {b_n} ) are ( a_1, a_4, a_5, a_6, a_9, a_{10} ), the next 6 terms are ( a_{11}, a_{14}, a_{15}, a_{16}, a_{19}, a_{20} ), and so on.Therefore, the ( k )-th block (where ( k ) starts at 0) contributes terms ( a_{10k + 1}, a_{10k + 4}, a_{10k + 5}, a_{10k + 6}, a_{10k + 9}, a_{10k + 10} ).So, each block contributes 6 terms, and there are 336 full blocks in 2016 terms (since 336 * 6 = 2016). Then, the 2017th term is the first term of the 337th block, and the 2018th term is the second term of the 337th block.Wait, let me check:Total number of terms up to block ( k ) is ( 6(k + 1) ). So, to find which block the 2018th term is in, we can compute ( 2018 / 6 ).2018 divided by 6 is 336 with a remainder of 2 (since 336 * 6 = 2016, and 2018 - 2016 = 2). So, the 2018th term is the 2nd term in the 337th block.Each block corresponds to a set of 10 consecutive ( n ) values. The first block is ( n = 1 ) to ( 10 ), the second block is ( n = 11 ) to ( 20 ), and so on. So, the 337th block corresponds to ( n = 10*336 + 1 = 3361 ) to ( n = 3370 ).Within each block, the terms are ordered as ( n = 10k + 1, 10k + 4, 10k + 5, 10k + 6, 10k + 9, 10k + 10 ). So, the first term in the block is ( n = 10k + 1 ), the second term is ( n = 10k + 4 ), and so on.Therefore, the 2nd term in the 337th block is ( n = 10*336 + 4 = 3364 ).So, ( b_{2018} = a_{3364} ).Now, I need to compute ( a_{3364} mod 7 ).First, let's compute ( a_n = n^3 - n mod 7 ).But instead of computing ( 3364^3 - 3364 ) directly, which is a huge number, I can compute ( 3364 mod 7 ) first, and then compute ( a_n mod 7 ) based on that.So, let's compute ( 3364 mod 7 ).To compute ( 3364 mod 7 ), I can divide 3364 by 7 and find the remainder.Alternatively, I can break down 3364:3364 divided by 7:7 * 480 = 3360, so 3364 - 3360 = 4.Therefore, ( 3364 mod 7 = 4 ).So, ( n equiv 4 mod 7 ).Therefore, ( a_n = n^3 - n equiv 4^3 - 4 mod 7 ).Compute ( 4^3 = 64 ). ( 64 mod 7 ): 7*9=63, so 64 - 63 = 1. Therefore, 64 ≡ 1 mod 7.So, ( 4^3 ≡ 1 mod 7 ).Then, ( 4^3 - 4 ≡ 1 - 4 = -3 mod 7 ).But -3 mod 7 is equivalent to 4, since 7 - 3 = 4.Therefore, ( a_{3364} equiv 4 mod 7 ).So, the remainder when ( b_{2018} ) is divided by 7 is 4.