💡First, I need to determine the unit's digit of each term in the product:1. For (41^{14}), the unit's digit of 41 is 1. Any number ending in 1 raised to any power will still have a unit's digit of 1.2. For (14^{87}), the unit's digit of 14 is 4. The powers of 4 cycle every two: 4, 6, 4, 6, etc. Since 87 is odd, the unit's digit is 4.3. For (87^{76}), the unit's digit of 87 is 7. The powers of 7 cycle every four: 7, 9, 3, 1, etc. Since 76 is a multiple of 4, the unit's digit is 1.Next, I multiply these unit's digits together: (1 times 4 times 1 = 4).Given that the unit's digit of the entire product is 4, the unit's digit contributed by the first term ((text{some number}^{41})) must be 1 to maintain the overall unit's digit of 4.Finally, to find the base of the first term, I consider that any number ending in 1 raised to any power will have a unit's digit of 1. Therefore, the base of the first term must have a unit's digit of 1.

💡To determine how many lerps embody a straight angle on Mars, we start by noting that a full circle is measured as 300 lerps. A straight angle, which is half of a full circle, can be calculated by dividing the total number of lerps by 2. So, we calculate:[frac{300 text{ lerps}}{2} = 150 text{ lerps}]Therefore, a straight angle is represented by 150 lerps.

💡Okay, so I have this equation here: ( x^2 - 6x - 5 = 0 ). I need to complete the square for this equation and figure out which of the given options it matches. Hmm, completing the square... I remember that's a method to solve quadratic equations by turning them into a perfect square trinomial, which can then be easily solved.Alright, let's start with the given equation:( x^2 - 6x - 5 = 0 )First, I think I need to move the constant term to the other side to get it ready for completing the square. So, I'll add 5 to both sides:( x^2 - 6x = 5 )Now, the next step is to create a perfect square trinomial on the left side. To do that, I need to take the coefficient of the x-term, which is -6, divide it by 2, and then square it. Let me calculate that:Coefficient of x: -6Divide by 2: -6 / 2 = -3Square it: (-3)^2 = 9Okay, so I need to add 9 to both sides of the equation to maintain the equality. Let's do that:( x^2 - 6x + 9 = 5 + 9 )Simplifying both sides:Left side: ( x^2 - 6x + 9 ) is a perfect square trinomial. It factors into ( (x - 3)^2 ).Right side: 5 + 9 = 14So now, the equation becomes:( (x - 3)^2 = 14 )Hmm, looking at the options provided:A: ( (x - 6)^2 = 30 )B: ( (x - 6)^2 = 41 )C: ( (x - 3)^2 = 4 )D: ( (x - 3)^2 = 14 )It looks like option D matches exactly with what I have here: ( (x - 3)^2 = 14 ). So, I think that's the correct answer.But just to make sure I didn't make any mistakes, let me double-check my steps.1. Original equation: ( x^2 - 6x - 5 = 0 )2. Move constant term: ( x^2 - 6x = 5 )3. Coefficient of x is -6. Half of that is -3, squared is 9.4. Add 9 to both sides: ( x^2 - 6x + 9 = 5 + 9 )5. Simplify: ( (x - 3)^2 = 14 )Yep, that seems correct. So, option D is definitely the right choice.I wonder why the other options are there. Maybe they're trying to trick me into thinking of different coefficients or different constants. For example, option A and B have ( (x - 6)^2 ), which would mean the coefficient was -12 instead of -6 because when you expand ( (x - 6)^2 ), you get ( x^2 - 12x + 36 ). But in our equation, the coefficient is -6, so that's not right.Option C has ( (x - 3)^2 = 4 ). That would mean after completing the square, the right side is 4, but in our case, it's 14. So, that's not correct either.Therefore, I'm confident that option D is the correct answer.

💡First, reduce the numbers modulo 50:- (289 equiv 39 pmod{50}) since 289 - 5*50 = 39- (673 equiv 23 pmod{50}) since 673 - 13*50 = 23Now, multiply these residues:[39 cdot 23 = 897]Next, figure out 897 modulo 50:- (897 equiv 47 pmod{50}) since 897 - 17*50 = 47 or just (897 div 50) leaves a remainder of 47.Thus,[289 cdot 673 equiv 47 pmod{50}]So, (m = boxed{47}).

💡Okay, so I need to figure out how many digits the number (8^{20}5^{18}) has when written out in base 10. Hmm, let's see. I remember that the number of digits in a number can be found using logarithms, specifically the base-10 logarithm. The formula is something like taking the floor of the log base 10 of the number and then adding 1. So, if I can find (log_{10}(8^{20}5^{18})), that should help me determine the number of digits.First, let me rewrite (8^{20}) in terms of base 2 because 8 is (2^3). So, (8^{20} = (2^3)^{20} = 2^{60}). That simplifies the expression a bit. Now, the original expression becomes (2^{60} times 5^{18}).I notice that (2^{60} times 5^{18}) can be rewritten by combining the powers of 2 and 5. Since (2 times 5 = 10), I can pair some of the 2s and 5s to make powers of 10. Specifically, I can take (5^{18}) and pair it with (2^{18}) from (2^{60}), which gives me (10^{18}). That leaves me with (2^{42}) because (60 - 18 = 42). So now, the expression is (10^{18} times 2^{42}).Now, (10^{18}) is a 1 followed by 18 zeros, which is a 19-digit number. But when I multiply it by (2^{42}), I need to figure out how that affects the number of digits. (2^{42}) is a specific number, so maybe I should calculate its approximate value or its logarithm to see how many digits it has.Let me calculate (log_{10}(2^{42})). Using logarithm properties, (log_{10}(2^{42}) = 42 times log_{10}(2)). I remember that (log_{10}(2)) is approximately 0.3010. So, (42 times 0.3010 = 12.642). That means (2^{42}) is approximately (10^{12.642}), which is a 13-digit number because the characteristic (the integer part) is 12, and we add 1 to get the number of digits.So, (10^{18}) is a 19-digit number, and (2^{42}) is a 13-digit number. When I multiply them together, I might think that the number of digits would just be the sum of the digits, but that's not quite right. Instead, the number of digits is determined by the sum of the exponents in the logarithm.Wait, let me think again. The total number of digits is given by (lfloor log_{10}(N) rfloor + 1), where (N) is the number. So, let's compute (log_{10}(8^{20}5^{18})).We already have (8^{20}5^{18} = 10^{18} times 2^{42}). Taking the logarithm, (log_{10}(10^{18} times 2^{42}) = log_{10}(10^{18}) + log_{10}(2^{42}) = 18 + 42 times log_{10}(2)). As before, (log_{10}(2) approx 0.3010), so (42 times 0.3010 = 12.642). Adding that to 18 gives (18 + 12.642 = 30.642).Now, taking the floor of 30.642 gives 30, and adding 1 gives 31. So, the number of digits should be 31.Wait, but earlier I thought (2^{42}) is a 13-digit number and (10^{18}) is a 19-digit number. If I multiply a 13-digit number by a 19-digit number, the result should have either 13 + 19 - 1 = 31 digits or 13 + 19 = 32 digits. But since (2^{42}) is less than (10^{13}), multiplying it by (10^{18}) would give a number less than (10^{31}), so the number of digits is 31.Yes, that makes sense. So, the number (8^{20}5^{18}) has 31 digits when written in base 10.

💡Okay, so I have these seven true or false questions about chemical equilibrium, and I need to figure out which ones are true and which ones are false. Let me take them one by one and think through each.Starting with question (1): "In the expression of the equilibrium constant, any concentration of substances can be used." Hmm, I remember that the equilibrium constant expression involves the concentrations of the products and reactants, each raised to the power of their stoichiometric coefficients. But wait, does that include all substances? I think solids and pure liquids are excluded because their concentrations are considered constant and don't affect the equilibrium. So, if the question says "any concentration," that might not be correct because solids and liquids aren't included. So, I think this statement is false.Moving on to question (2): "Catalysts can change the rate of chemical reactions and can also change the equilibrium constant." I know that catalysts speed up both the forward and reverse reactions equally, so they don't affect the position of equilibrium. But do they change the equilibrium constant? No, because the equilibrium constant depends only on temperature, not on the presence of a catalyst. So, this statement is false.Question (3): "If the equilibrium constant does not change, the chemical equilibrium definitely does not shift." Well, the equilibrium constant (K) is temperature-dependent. If K doesn't change, it means the temperature hasn't changed. However, even if K remains the same, the position of equilibrium can shift if other factors like concentration or pressure change. For example, adding more reactants would shift the equilibrium to produce more products, but K itself remains the same. So, this statement is false.Question (4): "For the reaction: 2SO₂(g) + O₂(g) ⇌ 2SO₃(g), if increasing the temperature decreases the value of K, then ΔH > 0." I recall that if increasing temperature decreases K, the reaction is exothermic, meaning ΔH is negative. Conversely, if increasing temperature increases K, the reaction is endothermic, and ΔH is positive. So, if K decreases with temperature, ΔH should be less than zero, not greater. Therefore, this statement is false.Question (5): "For a reversible reaction, if the stoichiometric numbers are different, the expressions and values of the chemical equilibrium constant are also different." The equilibrium constant expression depends on the balanced chemical equation, which includes stoichiometric coefficients. If the stoichiometric numbers change, the expression and the value of K would indeed change. So, this statement is true.Question (6): "When chemical equilibrium shifts, the equilibrium constant does not necessarily change." The equilibrium constant is only affected by temperature. If the shift in equilibrium is due to changes in concentration, pressure, or adding a catalyst (which doesn't affect K), then K remains the same. So, it's possible for equilibrium to shift without K changing. Therefore, this statement is true.Question (7): "Both the equilibrium constant and the conversion rate can reflect the extent to which a reversible reaction proceeds." The equilibrium constant indicates the ratio of products to reactants at equilibrium, showing the extent of the reaction. Conversion rate, or percent conversion, also shows how much reactant has been converted to product. So, both can reflect the extent of the reaction. This statement is true.Okay, I think I've gone through each question carefully. Now, I'll summarize my answers.