💡Alright, so I'm trying to figure out if Agent 007 can encrypt his number, which is 0.07, using two natural numbers m and n such that 0.07 equals 1/m plus 1/n. Okay, let's break this down.First, I know that natural numbers are positive integers like 1, 2, 3, and so on. So, m and n have to be positive integers. The equation given is:0.07 = 1/m + 1/nHmm, I need to find m and n such that when I add their reciprocals, I get 0.07. Maybe I can start by expressing 0.07 as a fraction to make it easier to work with. 0.07 is the same as 7/100. So, the equation becomes:7/100 = 1/m + 1/nNow, I need to find two fractions that add up to 7/100. Let me think about how to approach this. One way is to assume that m and n are different, so I can set up a system of equations. Let's say m is less than n for simplicity.So, 1/m + 1/n = 7/100I can rewrite this as:(n + m)/(m*n) = 7/100Cross-multiplying gives:100(n + m) = 7(m*n)Expanding that:100n + 100m = 7mnLet me rearrange this equation to make it easier to solve:7mn - 100m - 100n = 0Hmm, this looks like a quadratic in terms of m and n. Maybe I can factor it or find integer solutions by trial and error.Alternatively, I can express this as:7mn - 100m - 100n = 0Adding 10000 to both sides to complete the rectangle:7mn - 100m - 100n + 10000 = 10000Now, factor the left side:(7m - 100)(7n - 100) = 10000Wait, is that correct? Let me check:(7m - 100)(7n - 100) = 49mn - 700m - 700n + 10000But I have 7mn - 100m - 100n + 10000 = 10000So, it's not exactly matching. Maybe I need a different approach.Perhaps I can consider that 1/m and 1/n are both less than 0.07, so m and n must be greater than 1/0.07, which is approximately 14.2857. So, m and n must be at least 15.Let me try m = 20. Then 1/20 = 0.05. So, 0.07 - 0.05 = 0.02, which is 1/50. So, n would be 50.Let me check: 1/20 + 1/50 = 0.05 + 0.02 = 0.07. Yes, that works!So, m = 20 and n = 50.Okay, that seems straightforward. Now, for part b, Agent 013 wants to do the same with 0.13.So, 0.13 = 1/m + 1/nAgain, expressing 0.13 as a fraction: 13/100.So, 13/100 = 1/m + 1/nFollowing a similar approach:1/m + 1/n = 13/100(n + m)/(m*n) = 13/100Cross-multiplying:100(n + m) = 13(m*n)Expanding:100n + 100m = 13mnRearranging:13mn - 100m - 100n = 0Again, trying to factor or find integer solutions.Maybe I can try some values for m and see if n comes out as an integer.Let's try m = 8. Then 1/8 = 0.125. So, 0.13 - 0.125 = 0.005, which is 1/200. So, n would be 200.Checking: 1/8 + 1/200 = 0.125 + 0.005 = 0.13. Yes, that works!So, m = 8 and n = 200.Alternatively, I could have tried m = 200 and n = 8, but since m and n are interchangeable, both solutions are valid.So, Agent 013 can also encrypt his number using m = 8 and n = 200.I think that's it. Both agents can find natural numbers m and n to encrypt their respective numbers.

💡Okay, so I have this problem about an ellipse, and I need to find its focal length. The equation given is (frac{x^2}{20} + frac{y^2}{36} = 1). Hmm, I remember that ellipses have something to do with two foci, and the focal length is the distance between these two foci. But I'm a bit fuzzy on the exact steps to find it. Let me think.First, I recall that the standard form of an ellipse is (frac{(x-h)^2}{a^2} + frac{(y-k)^2}{b^2} = 1), where ((h,k)) is the center of the ellipse. In this case, the equation is (frac{x^2}{20} + frac{y^2}{36} = 1), so the center must be at the origin, (0,0), because there are no h or k terms. That makes things a bit simpler.Now, I need to figure out which one is (a^2) and which one is (b^2). I remember that in an ellipse, the larger denominator corresponds to the major axis, and the smaller one corresponds to the minor axis. So, comparing 20 and 36, 36 is larger. That means the major axis is along the y-axis because the larger denominator is under the y-term. Therefore, (a^2 = 36) and (b^2 = 20). Wait, let me make sure. I think (a) is always associated with the major axis, so if the major axis is along the y-axis, then (a^2 = 36), so (a = 6). Similarly, (b^2 = 20), so (b = sqrt{20}). Simplifying (sqrt{20}), that's (2sqrt{5}). Okay, so (a = 6) and (b = 2sqrt{5}).Now, how do I find the focal length? I remember that the distance from the center to each focus is denoted by (c), and the relationship between (a), (b), and (c) is given by (c^2 = a^2 - b^2). So, if I can find (c), then the focal length, which is the distance between the two foci, would be (2c).Let me plug in the values. (c^2 = a^2 - b^2 = 36 - 20 = 16). So, (c = sqrt{16} = 4). Therefore, the distance between the two foci is (2c = 2 times 4 = 8). Wait, let me double-check. If the major axis is along the y-axis, then the foci should be located at (0, ±c). So, plugging in, the foci would be at (0, 4) and (0, -4). The distance between these two points is indeed 8 units. That seems right.Just to recap, the steps I took were:1. Identify the standard form of the ellipse and determine the center.2. Determine which denominator is larger to identify the major and minor axes.3. Assign (a^2) and (b^2) accordingly.4. Use the relationship (c^2 = a^2 - b^2) to find (c).5. Multiply by 2 to get the focal length.I think that's all correct. I don't see any mistakes in my calculations, so I feel confident that the focal length is 8.**Final Answer**The focal length of the ellipse is boxed{8}.

💡First, I need to determine the merchant's cost price for the gadget. The gadget is acquired for 30 less 15%. To calculate this, I'll find 15% of 30 and subtract it from 30.Next, the merchant wants to sell the gadget at a gain of 40% on the cost price. I'll calculate 40% of the cost price and add it to the cost price to find the desired selling price.The merchant plans to offer a 25% discount on the marked price. Let's denote the marked price as ( x ). A 25% discount means the selling price will be 75% of the marked price. I'll set up the equation ( 0.75x ) equal to the desired selling price and solve for ( x ) to find the marked price.Finally, I'll compare the calculated marked price with the given options to determine the correct answer.

💡Okay, so I need to figure out how many positive divisors of 50! are prime. Hmm, let me start by recalling what a factorial is. 50! means 50 factorial, which is the product of all positive integers from 1 up to 50. So, 50! = 1 × 2 × 3 × ... × 50. Now, the question is about the positive divisors of 50! that are prime. I remember that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself. So, if I can find all the prime numbers that divide 50!, those will be the prime divisors we're looking for.Wait, but how do I find which primes divide 50!? Well, since 50! is the product of all numbers from 1 to 50, any prime number that is less than or equal to 50 will definitely be a factor of 50!. That makes sense because if a prime number is less than or equal to 50, it must appear in the multiplication sequence that makes up 50!.So, essentially, the prime divisors of 50! are all the prime numbers less than or equal to 50. Therefore, I need to list all the prime numbers up to 50 and count them. Let me try to list them out.Starting from the smallest prime, which is 2. Then 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Let me double-check to make sure I haven't missed any primes between 1 and 50.Wait, is 49 a prime? No, because 49 is 7 squared, so it's not prime. Similarly, 45 is divisible by 5 and 9, so not prime. 51 is beyond 50, so we don't need to consider that. Let me count the primes I have listed:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.That's 15 primes in total. Let me count them again to be sure: 2 (1), 3 (2), 5 (3), 7 (4), 11 (5), 13 (6), 17 (7), 19 (8), 23 (9), 29 (10), 31 (11), 37 (12), 41 (13), 43 (14), 47 (15). Yes, 15 primes.So, each of these primes divides 50! because they are all less than or equal to 50, and since 50! is the product of all integers up to 50, each prime must be a factor. Therefore, the number of positive prime divisors of 50! is 15.I think that's it. I don't see any primes less than or equal to 50 that I missed, and each of these primes is indeed a divisor of 50!. So, the answer should be 15.

💡First, I need to determine Joe's current weight based on his weight loss over the past 4 months and his projected weight in the next 3 months.I'll start by defining the variables:- Let ( W ) represent Joe's current weight.- Let ( R ) represent the average monthly weight loss rate.From the information given:1. Joe weighed 222 pounds 4 months ago, and his current weight is ( W ). This gives the equation: [ 222 - 4R = W ]2. In 3 months, Joe is expected to weigh 180 pounds. This gives the equation: [ W - 3R = 180 ]Next, I'll substitute the expression for ( W ) from the first equation into the second equation to solve for ( R ):[(222 - 4R) - 3R = 180]Simplifying this equation:[222 - 7R = 180][7R = 222 - 180][7R = 42][R = 6]Now that I know the monthly weight loss rate is 6 pounds, I'll substitute ( R ) back into the first equation to find Joe's current weight:[W = 222 - 4(6)][W = 222 - 24][W = 198]Therefore, Joe's current weight is 198 pounds.

💡Alright, so I've got this equation to solve: tan(2x) + tan(5x) = sec(5x). Hmm, okay. I remember that tan and sec are trigonometric functions, and they have some identities that might help here. Let me think about how to approach this.First, maybe I can rewrite everything in terms of sine and cosine to simplify. Let's see, tan(2x) is sin(2x)/cos(2x), tan(5x) is sin(5x)/cos(5x), and sec(5x) is 1/cos(5x). So, substituting these in, the equation becomes:sin(2x)/cos(2x) + sin(5x)/cos(5x) = 1/cos(5x)Hmm, okay. Let me write that down:sin(2x)/cos(2x) + sin(5x)/cos(5x) = 1/cos(5x)Looking at this, I notice that on the right side, we have 1/cos(5x), and on the left side, there's a sin(5x)/cos(5x). Maybe I can subtract sin(5x)/cos(5x) from both sides to isolate sin(2x)/cos(2x). Let's try that:sin(2x)/cos(2x) = 1/cos(5x) - sin(5x)/cos(5x)Simplify the right side:1/cos(5x) - sin(5x)/cos(5x) = (1 - sin(5x))/cos(5x)So now the equation is:sin(2x)/cos(2x) = (1 - sin(5x))/cos(5x)Hmm, okay. So, tan(2x) = (1 - sin(5x))/cos(5x). Maybe I can find a way to express (1 - sin(5x))/cos(5x) in terms of a tangent function as well. I remember there's an identity for tan(theta/2) which is (1 - cos(theta))/sin(theta). Wait, maybe I can manipulate (1 - sin(5x))/cos(5x) to look like that.Let me think. If I consider 1 - sin(5x), maybe I can relate it to cos of some angle. I recall that sin(theta) = cos(pi/2 - theta). So, sin(5x) = cos(pi/2 - 5x). Therefore, 1 - sin(5x) = 1 - cos(pi/2 - 5x). So, substituting back, (1 - sin(5x))/cos(5x) becomes (1 - cos(pi/2 - 5x))/cos(5x). Now, using the identity tan(theta/2) = (1 - cos(theta))/sin(theta), if I let theta = pi/2 - 5x, then:tan((pi/2 - 5x)/2) = (1 - cos(pi/2 - 5x))/sin(pi/2 - 5x)But sin(pi/2 - 5x) is cos(5x), right? So, substituting that in, we get:tan((pi/4 - (5x)/2)) = (1 - cos(pi/2 - 5x))/cos(5x)Which is exactly the numerator and denominator we have. So, (1 - sin(5x))/cos(5x) is equal to tan(pi/4 - (5x)/2). Therefore, our equation becomes:tan(2x) = tan(pi/4 - (5x)/2)Okay, so now we have tan(A) = tan(B), which implies that A = B + n*pi for some integer n, because the tangent function has a period of pi.So, setting 2x = pi/4 - (5x)/2 + n*pi.Let me write that equation:2x = pi/4 - (5x)/2 + n*piNow, let's solve for x. First, let's get all the x terms on one side. So, add (5x)/2 to both sides:2x + (5x)/2 = pi/4 + n*piTo combine the x terms, let's convert 2x to 4x/2:4x/2 + 5x/2 = 9x/2So, 9x/2 = pi/4 + n*piNow, solve for x by multiplying both sides by 2/9:x = (2/9)(pi/4 + n*pi) = (2/9)(pi/4) + (2/9)(n*pi) = pi/18 + (2n pi)/9So, x = pi/18 + (2n pi)/9Now, we need to find the smallest positive solution. Let's consider n = 0:x = pi/18 + 0 = pi/18Is this a valid solution? Let's check if it satisfies the original equation.Compute tan(2x) + tan(5x) and see if it equals sec(5x).First, x = pi/18.Compute 2x = 2*(pi/18) = pi/9Compute 5x = 5*(pi/18) = 5pi/18So, tan(pi/9) + tan(5pi/18) = sec(5pi/18)Let me compute each term numerically to check.First, pi is approximately 3.1416.So, pi/9 ≈ 0.3491 radians5pi/18 ≈ 0.8727 radiansCompute tan(pi/9):tan(0.3491) ≈ 0.3640Compute tan(5pi/18):tan(0.8727) ≈ 1.2090Compute sec(5pi/18):sec(0.8727) = 1/cos(0.8727) ≈ 1/0.6428 ≈ 1.5557Now, tan(pi/9) + tan(5pi/18) ≈ 0.3640 + 1.2090 ≈ 1.5730Compare to sec(5pi/18) ≈ 1.5557Hmm, these are close but not exactly equal. Maybe due to rounding errors? Let me check with more precise calculations.Alternatively, perhaps there's a mistake in the manipulation. Let me double-check.Starting from tan(2x) = tan(pi/4 - (5x)/2)So, 2x = pi/4 - (5x)/2 + n*piMultiply both sides by 2 to eliminate denominators:4x = pi/2 - 5x + 2n piBring 5x to the left:4x + 5x = pi/2 + 2n pi9x = pi/2 + 2n piSo, x = (pi/2 + 2n pi)/9 = pi/(18) + (2n pi)/9Wait, that's the same as before. So, x = pi/18 + (2n pi)/9So, for n=0, x=pi/18≈0.1745 radiansWait, but earlier when I plugged in x=pi/18, I got tan(2x)+tan(5x)≈1.5730 and sec(5x)≈1.5557, which are close but not exactly equal. Maybe due to calculator precision? Let me compute more accurately.Compute tan(pi/9):pi/9≈0.3490658504 radianstan(pi/9)=tan(20 degrees)= approximately 0.3640tan(5pi/18)=tan(50 degrees)= approximately 1.191753592Wait, earlier I said 1.2090, which is incorrect. Let me correct that.tan(50 degrees)=tan(5pi/18)= approximately 1.191753592So, tan(pi/9)+tan(5pi/18)=0.3640 + 1.19175≈1.55575And sec(5pi/18)=1/cos(5pi/18)=1/cos(50 degrees)=1/0.6427876097≈1.555723205So, 1.55575 vs 1.555723205. That's very close, with the slight difference due to rounding. So, x=pi/18 is indeed a solution.Therefore, the smallest positive solution is pi/18.Wait, but let me check for n=-1. If n=-1, x=pi/18 - 2pi/9=pi/18 - 4pi/18= -3pi/18=-pi/6, which is negative, so not positive. So, the smallest positive solution is pi/18.Thus, the answer is pi/18.