💡First, I need to determine the speeds of both bullet trains. Since each train takes a specific amount of time to cross a telegraph post, I can use the formula:[text{Speed} = frac{text{Distance}}{text{Time}}]For the first train, which takes 8 seconds to cross the telegraph post:[V_1 = frac{120 text{ meters}}{8 text{ seconds}} = 15 text{ meters/second}]For the second train, which takes 15 seconds to cross the telegraph post:[V_2 = frac{120 text{ meters}}{15 text{ seconds}} = 8 text{ meters/second}]When the trains are moving in opposite directions, their relative speed is the sum of their individual speeds:[V_r = V_1 + V_2 = 15 text{ meters/second} + 8 text{ meters/second} = 23 text{ meters/second}]The total distance to be covered when they cross each other is the sum of their lengths:[D = 120 text{ meters} + 120 text{ meters} = 240 text{ meters}]Finally, the time it takes for them to cross each other is calculated by dividing the total distance by the relative speed:[T = frac{D}{V_r} = frac{240 text{ meters}}{23 text{ meters/second}} approx 10.43 text{ seconds}]

💡Okay, so I have this problem here: I need to find the least positive integer value of ( x ) such that ( (3x)^2 + 2 cdot 58 cdot 3x + 58^2 ) is a multiple of 53. Hmm, that sounds a bit complicated, but maybe I can break it down step by step.First, let me look at the expression: ( (3x)^2 + 2 cdot 58 cdot 3x + 58^2 ). Wait a minute, that looks familiar. Isn't that similar to the expansion of a binomial squared? Let me check: ( (a + b)^2 = a^2 + 2ab + b^2 ). Yeah, so if I let ( a = 3x ) and ( b = 58 ), then the expression becomes ( (3x + 58)^2 ). So, the problem simplifies to finding the smallest positive integer ( x ) such that ( (3x + 58)^2 ) is a multiple of 53.Okay, so ( (3x + 58)^2 ) being a multiple of 53 means that 53 divides ( (3x + 58)^2 ). Now, I remember that if a prime number divides a square, then it must divide the base as well. Since 53 is a prime number, this should apply here. So, 53 must divide ( 3x + 58 ). That means ( 3x + 58 ) is a multiple of 53. Let me write that down: ( 3x + 58 equiv 0 mod{53} ). So, I need to solve this congruence for ( x ). Let me rearrange it: ( 3x equiv -58 mod{53} ). Hmm, negative numbers in modular arithmetic can be a bit tricky, but I can convert -58 to a positive equivalent modulo 53.To do that, I can add 53 to -58 until I get a positive number. Let's see: ( -58 + 53 = -5 ). That's still negative. Adding another 53: ( -5 + 53 = 48 ). Okay, so ( -58 equiv 48 mod{53} ). So, now the equation becomes ( 3x equiv 48 mod{53} ).Now, I need to solve for ( x ). This is a linear congruence of the form ( ax equiv b mod{m} ), where ( a = 3 ), ( b = 48 ), and ( m = 53 ). To solve this, I need to find the multiplicative inverse of 3 modulo 53. The inverse of 3 is a number ( y ) such that ( 3y equiv 1 mod{53} ).Let me try to find this inverse. I can use the Extended Euclidean Algorithm for this. The algorithm finds integers ( y ) and ( k ) such that ( 3y + 53k = 1 ). Let's apply the algorithm step by step.First, divide 53 by 3:- 53 divided by 3 is 17 with a remainder of 2. So, ( 53 = 3 times 17 + 2 ).Next, divide 3 by the remainder 2:- 3 divided by 2 is 1 with a remainder of 1. So, ( 3 = 2 times 1 + 1 ).Now, divide 2 by the remainder 1:- 2 divided by 1 is 2 with a remainder of 0. So, the GCD is 1, which means the inverse exists.Now, let's backtrack to express 1 as a combination of 3 and 53:From the second step: ( 1 = 3 - 2 times 1 ).But from the first step, ( 2 = 53 - 3 times 17 ). Substitute this into the equation:( 1 = 3 - (53 - 3 times 17) times 1 ).Simplify:( 1 = 3 - 53 + 3 times 17 ).Combine like terms:( 1 = 3 times (1 + 17) - 53 times 1 ).( 1 = 3 times 18 - 53 times 1 ).So, the multiplicative inverse of 3 modulo 53 is 18 because ( 3 times 18 = 54 equiv 1 mod{53} ).Now, going back to the congruence ( 3x equiv 48 mod{53} ), I can multiply both sides by 18 to solve for ( x ):( x equiv 48 times 18 mod{53} ).Let me calculate ( 48 times 18 ):First, 48 times 10 is 480, and 48 times 8 is 384. So, 480 + 384 = 864.Now, I need to find 864 modulo 53. Let's divide 864 by 53 to find the remainder.53 times 16 is 848 because 53 times 10 is 530, 53 times 6 is 318, so 530 + 318 = 848.Subtracting that from 864: 864 - 848 = 16.So, ( 864 mod{53} = 16 ). Therefore, ( x equiv 16 mod{53} ).Since we're looking for the least positive integer value of ( x ), the smallest positive solution is 16.Wait, let me double-check my calculations to make sure I didn't make a mistake. So, starting from the beginning: the expression simplifies to ( (3x + 58)^2 ), which needs to be divisible by 53. Therefore, ( 3x + 58 ) must be divisible by 53. So, ( 3x + 58 equiv 0 mod{53} ), which simplifies to ( 3x equiv -58 mod{53} ). Converting -58 to a positive equivalent modulo 53 gives 48, so ( 3x equiv 48 mod{53} ). Finding the inverse of 3 modulo 53 gives 18, so multiplying both sides by 18 gives ( x equiv 48 times 18 mod{53} ). Calculating 48 times 18 gives 864, and 864 divided by 53 is 16 with a remainder of 16. So, yes, ( x equiv 16 mod{53} ), meaning the smallest positive integer ( x ) is 16.I think that makes sense. I don't see any errors in my steps, so I'm pretty confident that 16 is the correct answer.

💡Alright, so I have this problem where I need to find the greatest common divisor (GCD) of ( g(50) ) and ( g(53) ) where the function ( g(x) ) is defined as ( x^2 - 3x + 2025 ). Hmm, okay, let me break this down step by step.First, I think I should compute the values of ( g(50) ) and ( g(53) ) individually. That way, I can have concrete numbers to work with when calculating their GCD. Let me write out the function again to make sure I have it right:( g(x) = x^2 - 3x + 2025 )So, for ( g(50) ), I substitute 50 into the function:( g(50) = (50)^2 - 3(50) + 2025 )Calculating each term:- ( 50^2 = 2500 )- ( 3 times 50 = 150 )So, plugging those in:( g(50) = 2500 - 150 + 2025 )Let me do the subtraction first:2500 - 150 = 2350Then add 2025:2350 + 2025 = 4375Okay, so ( g(50) = 4375 ).Now, let's compute ( g(53) ):( g(53) = (53)^2 - 3(53) + 2025 )Calculating each term:- ( 53^2 = 2809 )- ( 3 times 53 = 159 )So, plugging those in:( g(53) = 2809 - 159 + 2025 )First, subtract 159 from 2809:2809 - 159 = 2650Then add 2025:2650 + 2025 = 4675So, ( g(53) = 4675 ).Now, I need to find the GCD of 4375 and 4675. I remember that the Euclidean algorithm is a reliable method for finding the GCD of two numbers. Let me recall how that works.The Euclidean algorithm involves dividing the larger number by the smaller one and then replacing the larger number with the smaller number and the smaller number with the remainder from the division. This process repeats until the remainder is zero, and the non-zero remainder just before that is the GCD.So, let's apply that here.First, divide 4675 by 4375.4675 ÷ 4375 = 1 with a remainder. Let me calculate the remainder:4675 - (4375 × 1) = 4675 - 4375 = 300So, the remainder is 300. Now, I replace the numbers: the new pair is 4375 and 300.Next step: divide 4375 by 300.4375 ÷ 300 = 14 with a remainder. Let me compute that:300 × 14 = 4200Subtract that from 4375:4375 - 4200 = 175So, the remainder is 175. Now, the pair is 300 and 175.Divide 300 by 175:300 ÷ 175 = 1 with a remainder.175 × 1 = 175Subtract from 300:300 - 175 = 125Remainder is 125. Now, the pair is 175 and 125.Divide 175 by 125:175 ÷ 125 = 1 with a remainder.125 × 1 = 125Subtract from 175:175 - 125 = 50Remainder is 50. Now, the pair is 125 and 50.Divide 125 by 50:125 ÷ 50 = 2 with a remainder.50 × 2 = 100Subtract from 125:125 - 100 = 25Remainder is 25. Now, the pair is 50 and 25.Divide 50 by 25:50 ÷ 25 = 2 with a remainder of 0.Since the remainder is now 0, the algorithm stops, and the last non-zero remainder is 25. Therefore, the GCD of 4375 and 4675 is 25.Wait a second, but I think I might have made a mistake here because when I did the initial calculations, I thought the GCD was 75. Let me double-check my steps.Wait, no, actually, in my initial calculation, I think I might have confused the steps. Let me go through the Euclidean algorithm again more carefully.Starting with 4675 and 4375:4675 ÷ 4375 = 1 with a remainder of 300.So, GCD(4675, 4375) = GCD(4375, 300).Now, 4375 ÷ 300 = 14 with a remainder of 175.So, GCD(4375, 300) = GCD(300, 175).300 ÷ 175 = 1 with a remainder of 125.GCD(300, 175) = GCD(175, 125).175 ÷ 125 = 1 with a remainder of 50.GCD(175, 125) = GCD(125, 50).125 ÷ 50 = 2 with a remainder of 25.GCD(125, 50) = GCD(50, 25).50 ÷ 25 = 2 with a remainder of 0.So, GCD is 25.Wait, that contradicts my initial thought of 75. Did I make a mistake earlier?Let me check the calculations again.Compute ( g(50) ):50^2 = 25003*50 = 1502500 - 150 = 23502350 + 2025 = 4375. That seems correct.Compute ( g(53) ):53^2 = 28093*53 = 1592809 - 159 = 26502650 + 2025 = 4675. That also seems correct.Now, applying the Euclidean algorithm:GCD(4675, 4375)4675 - 4375 = 300GCD(4375, 300)4375 ÷ 300 = 14, remainder 4375 - 300*14 = 4375 - 4200 = 175GCD(300, 175)300 - 175 = 125GCD(175, 125)175 - 125 = 50GCD(125, 50)125 - 2*50 = 25GCD(50,25)50 ÷25=2, remainder 0.So, GCD is 25.Wait, but in the initial problem, the assistant's answer was 75. Did I make a mistake in the calculations?Wait, let me check the function again. Maybe I misread it.The function is ( g(x) = x^2 - 3x + 2025 ). Yes, that's correct.Calculating ( g(50) ):50^2 = 25003*50 = 1502500 - 150 = 23502350 + 2025 = 4375. Correct.( g(53) ):53^2 = 28093*53 = 1592809 - 159 = 26502650 + 2025 = 4675. Correct.So, the numbers are correct. Then why did the assistant get 75?Wait, perhaps I made a mistake in the Euclidean algorithm steps.Let me try a different approach. Maybe factorizing both numbers.Factorizing 4375:4375 ÷ 5 = 875875 ÷ 5 = 175175 ÷ 5 = 3535 ÷ 5 = 7So, 4375 = 5^4 * 7Factorizing 4675:4675 ÷ 5 = 935935 ÷ 5 = 187187 ÷ 11 = 17So, 4675 = 5^2 * 11 * 17Now, the common prime factors are 5^2, which is 25.So, GCD is 25.Wait, that's consistent with my earlier result. So, the GCD should be 25, not 75.But in the initial problem, the assistant's answer was 75. Did I misinterpret something?Wait, perhaps the function was different? Let me check again.The function is ( g(x) = x^2 - 3x + 2025 ). Yes, that's correct.Wait, maybe I made a mistake in the initial calculation of ( g(50) ) and ( g(53) ).Let me recalculate:( g(50) = 50^2 - 3*50 + 2025 = 2500 - 150 + 2025 = 2500 - 150 = 2350 + 2025 = 4375 ). Correct.( g(53) = 53^2 - 3*53 + 2025 = 2809 - 159 + 2025 = 2809 - 159 = 2650 + 2025 = 4675 ). Correct.So, the numbers are correct. Then why is the GCD 25 and not 75?Wait, 75 is 3*5^2. But in the factorization of 4375, we have 5^4 *7, and in 4675, we have 5^2 *11*17. So, the common factors are only 5^2, which is 25. There is no factor of 3 in both numbers.Wait, but 4375 is 5^4 *7, and 4675 is 5^2 *11*17. So, 75 is 3*5^2, but 3 is not a common factor. So, GCD is 25.Therefore, the correct GCD is 25, not 75.Wait, but in the initial problem, the assistant's answer was 75. Did they make a mistake?Alternatively, perhaps I made a mistake in the function. Let me check again.The function is ( g(x) = x^2 - 3x + 2025 ). Yes, that's correct.Wait, maybe I should try another approach. Let me consider the difference between ( g(53) ) and ( g(50) ).( g(53) - g(50) = (53^2 - 3*53 + 2025) - (50^2 - 3*50 + 2025) )Simplify:= (53^2 - 50^2) - 3(53 - 50)= (53 - 50)(53 + 50) - 3*3= 3*103 - 9= 309 - 9 = 300So, the difference is 300. Therefore, GCD(g(53), g(50)) must divide 300.But we already know that GCD is 25, which divides 300.Alternatively, perhaps I can use the property that GCD(g(a), g(b)) divides g(a) - g(b). So, GCD(g(53), g(50)) divides 300.But since we found that GCD is 25, which divides 300, that's consistent.Wait, but 75 also divides 300. So, why isn't 75 the GCD?Because when we factorize both numbers, 4375 and 4675, their common factors are only 25. So, even though 75 divides 300, it doesn't divide both 4375 and 4675.Let me check:Does 75 divide 4375?4375 ÷ 75 = 58.333... So, no, it doesn't divide evenly.Similarly, 4675 ÷ 75 = 62.333... So, no.Therefore, 75 is not a common divisor.Hence, the correct GCD is 25.Wait, but in the initial problem, the assistant's answer was 75. Maybe they made a mistake in their calculation.Alternatively, perhaps I made a mistake in my factorization.Let me double-check the factorization of 4375 and 4675.4375:Divide by 5: 4375 ÷5=875875 ÷5=175175 ÷5=3535 ÷5=7So, 4375=5^4 *74675:4675 ÷5=935935 ÷5=187187 ÷11=17So, 4675=5^2 *11*17Yes, that's correct. So, common factors are 5^2=25.Therefore, GCD is 25.So, I think the initial assistant's answer of 75 was incorrect. The correct GCD is 25.

💡First, I identify the cost of a bus ride from town P to town Q, which is 3.75.Next, I determine the cost of a train ride by adding the additional amount to the bus ride cost: 3.75 + 2.35 = 6.10.Finally, I calculate the total cost of one train ride and one bus ride by adding the two amounts together: 6.10 + 3.75 = 9.85.

💡Okay, so I need to figure out all the values of ( b ) such that the two given lines are skew. Hmm, skew lines... I remember that skew lines are lines that don't intersect and aren't parallel. So, they have to be non-parallel and non-intersecting. First, let me write down the equations of the lines to get a clear picture.The first line is given by:[begin{pmatrix} 2 3 b end{pmatrix} + t begin{pmatrix} 3 4 5 end{pmatrix}]So, in parametric form, this can be written as:[x = 2 + 3t, quad y = 3 + 4t, quad z = b + 5t]The second line is:[begin{pmatrix} 5 3 1 end{pmatrix} + u begin{pmatrix} 7 3 2 end{pmatrix}]Which translates to:[x = 5 + 7u, quad y = 3 + 3u, quad z = 1 + 2u]Alright, so to check if these lines are skew, I need to ensure two things: they don't intersect and they aren't parallel. Let me first check if they are parallel. Looking at the direction vectors:- First line: ( begin{pmatrix} 3 4 5 end{pmatrix} )- Second line: ( begin{pmatrix} 7 3 2 end{pmatrix} )Are these scalar multiples of each other? Let's see. If there exists a scalar ( k ) such that ( 3 = 7k ), ( 4 = 3k ), and ( 5 = 2k ), then they are parallel. From the first component: ( k = 3/7 ). Let's check the second component: ( 4 = 3*(3/7) = 9/7 ). But 4 is not equal to 9/7. So, they aren't parallel. Good, that means they could potentially be skew.Now, I need to check if they intersect. If they don't intersect, then they are skew. So, let's set their parametric equations equal to each other and see if there's a solution for ( t ) and ( u ).Setting the x, y, z components equal:1. ( 2 + 3t = 5 + 7u )2. ( 3 + 4t = 3 + 3u )3. ( b + 5t = 1 + 2u )Let me solve these equations step by step.Starting with equation 2:( 3 + 4t = 3 + 3u )Subtract 3 from both sides:( 4t = 3u )So, ( t = (3/4)u )Now, substitute ( t = (3/4)u ) into equation 1:( 2 + 3*(3/4)u = 5 + 7u )Simplify:( 2 + (9/4)u = 5 + 7u )Subtract 2 from both sides:( (9/4)u = 3 + 7u )Multiply both sides by 4 to eliminate the fraction:( 9u = 12 + 28u )Subtract 9u from both sides:( 0 = 12 + 19u )So, ( 19u = -12 )Thus, ( u = -12/19 )Now, substitute ( u = -12/19 ) back into ( t = (3/4)u ):( t = (3/4)*(-12/19) = (-36)/76 = (-9)/19 )So, we have ( t = -9/19 ) and ( u = -12/19 ). Now, let's check if these values satisfy equation 3.Equation 3:( b + 5t = 1 + 2u )Substitute ( t = -9/19 ) and ( u = -12/19 ):( b + 5*(-9/19) = 1 + 2*(-12/19) )Simplify:( b - 45/19 = 1 - 24/19 )Convert 1 to 19/19:( b - 45/19 = 19/19 - 24/19 )( b - 45/19 = -5/19 )Add 45/19 to both sides:( b = (-5/19) + 45/19 = 40/19 )So, if ( b = 40/19 ), the lines intersect. Therefore, for the lines to be skew, ( b ) must not equal ( 40/19 ). Wait, let me just double-check my calculations to make sure I didn't make any mistakes.Starting with equation 2: ( 4t = 3u ) leading to ( t = (3/4)u ). Plugging into equation 1: ( 2 + 3*(3/4)u = 5 + 7u ) simplifies correctly to ( 2 + (9/4)u = 5 + 7u ). Subtracting 2: ( (9/4)u = 3 + 7u ). Multiply by 4: ( 9u = 12 + 28u ). Subtract 9u: ( 0 = 12 + 19u ), so ( u = -12/19 ). Then ( t = (3/4)*(-12/19) = -9/19 ). Plugging into equation 3: ( b + 5*(-9/19) = 1 + 2*(-12/19) ). That gives ( b - 45/19 = 1 - 24/19 ). 1 is 19/19, so ( b - 45/19 = -5/19 ). Adding 45/19: ( b = 40/19 ). Yep, seems correct.Therefore, the lines intersect when ( b = 40/19 ). So, for them to be skew, ( b ) must be any real number except ( 40/19 ).I think that's it. So, the answer is all real numbers except ( 40/19 ).**Final Answer**The lines are skew for all real numbers ( b ) except ( frac{40}{19} ). Thus, the values of ( b ) are ( boxed{left(-infty, frac{40}{19}right) cup left(frac{40}{19}, inftyright)} ).**Final Answer**The lines are skew for all real numbers ( b ) except ( frac{40}{19} ). Thus, the values of ( b ) are ( boxed{left(-infty, frac{40}{19}right) cup left(frac{40}{19}, inftyright)} ).

💡First, I need to identify the multiples of 11 between 100 and 200. These numbers are 110, 121, 132, 143, 154, 165, 176, 187, and 198.Next, I should eliminate the numbers that are multiples of 2, as the preferred number is not even. This removes 110, 132, 154, 176, and 198 from consideration.Now, I have the remaining numbers: 121, 143, 165, and 187. I need to check if the sum of their digits is a multiple of 3.For 121: 1 + 2 + 1 = 4. 4 is not a multiple of 3.For 143: 1 + 4 + 3 = 8. 8 is not a multiple of 3.For 165: 1 + 6 + 5 = 12. 12 is a multiple of 3.For 187: 1 + 8 + 7 = 16. 16 is not a multiple of 3.The only number that meets all the criteria is 165.