💡Okay, so I need to find the remainder when 444 raised to the power of 444 is divided by 13. Hmm, that seems pretty big. I remember that when dealing with remainders and exponents, modular arithmetic can be really helpful. Let me try to recall how that works.First, I think I should simplify 444 modulo 13. That means I need to find what 444 is equivalent to when divided by 13. So, let me divide 444 by 13 and find the remainder. Let me do the division: 13 times 34 is 442, right? Because 13 times 30 is 390, and 13 times 4 is 52, so 390 plus 52 is 442. So, 444 minus 442 is 2. Wait, no, that can't be right because 444 divided by 13 is 34 with a remainder of 2? Wait, hold on, 13 times 34 is 442, so 444 minus 442 is 2. So, 444 is congruent to 2 modulo 13. Wait, but earlier I thought it was 4. Did I make a mistake?Let me double-check. 13 times 34 is 442, correct. So, 444 minus 442 is 2. So, 444 ≡ 2 mod 13. Hmm, maybe I confused it with another number earlier. So, 444 is actually 2 mod 13, not 4. Okay, that's important.So, now, the problem becomes finding 2^444 mod 13. That's still a huge exponent, but maybe there's a pattern or a cycle in the powers of 2 modulo 13 that I can use.I remember something called Fermat's Little Theorem, which says that if p is a prime number, then for any integer a not divisible by p, a^(p-1) ≡ 1 mod p. Since 13 is a prime number, and 2 is not divisible by 13, this should apply here. So, 2^12 ≡ 1 mod 13.That's useful because 444 divided by 12 will give me how many full cycles of 12 there are in the exponent, and then I can find the remainder. Let me calculate 444 divided by 12.444 divided by 12 is 37 with a remainder of 0, right? Because 12 times 37 is 444 exactly. So, 444 = 12 * 37. Therefore, 2^444 = (2^12)^37. Since 2^12 ≡ 1 mod 13, this becomes 1^37 mod 13, which is just 1.Wait, so does that mean the remainder is 1? Let me make sure I didn't skip any steps.So, 444 ≡ 2 mod 13, so 444^444 ≡ 2^444 mod 13. Then, using Fermat's Little Theorem, since 13 is prime and 2 is not a multiple of 13, 2^12 ≡ 1 mod 13. Therefore, 2^444 = (2^12)^37 ≡ 1^37 ≡ 1 mod 13. Yeah, that seems right.But just to be thorough, maybe I should check the cycle of powers of 2 modulo 13 to see if there's a shorter cycle than 12. Let me compute the powers of 2 modulo 13:2^1 = 2 mod 132^2 = 4 mod 132^3 = 8 mod 132^4 = 16 ≡ 3 mod 132^5 = 6 mod 132^6 = 12 mod 132^7 = 24 ≡ 11 mod 132^8 = 22 ≡ 9 mod 132^9 = 18 ≡ 5 mod 132^10 = 10 mod 132^11 = 20 ≡ 7 mod 132^12 = 14 ≡ 1 mod 13Okay, so the cycle length here is indeed 12, which matches Fermat's Little Theorem. So, every 12 exponents, it cycles back to 1. Therefore, 2^444 is (2^12)^37, which is 1^37, which is 1. So, the remainder is 1.Wait, but earlier I thought 444 was congruent to 4 mod 13, but then I realized it was 2. Did I make a mistake in the initial step? Let me check again.444 divided by 13: 13*34=442, so 444-442=2. So, yes, 444 ≡ 2 mod 13. So, 444^444 ≡ 2^444 mod 13. Then, using Fermat's Little Theorem, since 2^12 ≡ 1 mod 13, and 444 is a multiple of 12, 2^444 ≡ 1 mod 13. So, the remainder is 1.I think that's correct. Maybe I can try a smaller exponent to test. For example, 2^12 ≡ 1 mod 13, so 2^12 = 4096. 4096 divided by 13: 13*315=4095, so 4096 ≡ 1 mod 13. Yep, that works. So, 2^12 ≡ 1, and since 444 is 12*37, 2^444 is (2^12)^37 ≡ 1^37 ≡ 1 mod 13.Therefore, the remainder when 444^444 is divided by 13 is 1.

💡Okay, so I have this problem where I'm given an odd function f(x). For x less than zero, f(x) is defined as x squared plus x. I need to find the expression for f(x) when x is greater than zero. Hmm, let's see. First, I remember that an odd function has a specific property. It's something like f of negative x equals negative f of x. So, if I plug in a negative value into the function, it should be the same as taking the negative of the function evaluated at the positive value. That makes sense because odd functions are symmetric about the origin, right? So, if I know the function for negative x, I can find it for positive x by using this property.Alright, so for x less than zero, f(x) is x squared plus x. Let me write that down: f(x) = x² + x when x < 0. Now, I need to find f(x) when x > 0. Since the function is odd, I can use the property f(-x) = -f(x). Let me think about how to apply this. If I take a positive x, say x = a where a > 0, then -a is less than zero. So, f(-a) would be (-a)² + (-a), right? Let me compute that. (-a) squared is a squared, and then plus (-a) is just minus a. So, f(-a) = a² - a. But since f is odd, f(-a) should also equal -f(a). So, I can set up the equation: a² - a = -f(a). To solve for f(a), I can multiply both sides by -1. That gives me f(a) = -a² + a. So, f(a) is equal to negative a squared plus a. Since a was just a positive number, I can replace a with x to make it more general. So, f(x) = -x² + x for x > 0. Wait, let me double-check that. If I plug in a negative x into f(x), I get x squared plus x. Then, using the odd function property, f(-x) should be -f(x). So, f(-x) = x² - x, which is equal to -f(x). So, f(x) must be -x² + x. Yeah, that seems consistent. Let me also think about the graph of this function. For x < 0, it's a parabola opening upwards because of the x squared term, and then it has a linear term x. For x > 0, it's a downward opening parabola because of the negative x squared, and then a linear term x. So, the graph should be symmetric about the origin, which is what we expect for an odd function. Is there another way to verify this? Maybe by plugging in specific values. Let's say x = 1. For x = 1, f(1) should be -1² + 1, which is -1 + 1 = 0. Now, let's check f(-1). According to the given function, f(-1) = (-1)² + (-1) = 1 - 1 = 0. So, f(-1) = 0, and -f(1) should also be 0. That works. Let me try another value, like x = 2. f(2) would be -2² + 2 = -4 + 2 = -2. Then, f(-2) should be (-2)² + (-2) = 4 - 2 = 2. And -f(2) is -(-2) = 2. So, that also checks out. How about x = -3? f(-3) is (-3)² + (-3) = 9 - 3 = 6. Then, f(3) should be -3² + 3 = -9 + 3 = -6. And -f(3) is -(-6) = 6, which matches f(-3). Okay, so plugging in these specific values seems to confirm that the function is odd and that the expression for x > 0 is indeed -x² + x. I think I've got it. The key was remembering the definition of an odd function and applying it correctly. It's important to substitute -x into the given function and then solve for f(x) using the property f(-x) = -f(x). Also, verifying with specific values helped solidify that the derived expression is correct.

💡Okay, so I have this problem where I'm given that the tangent of alpha is -2, and I need to find the value of this expression: (sin2α - cos²α)/sin²α. Hmm, let me think about how to approach this step by step.First, I remember that tanα is equal to sinα over cosα, so if tanα is -2, that means sinα/cosα = -2. Maybe I can express sinα and cosα in terms of each other. Let me denote sinα as -2k and cosα as k, where k is some constant. But wait, I should remember that sin²α + cos²α = 1, so substituting those in, I get (-2k)² + (k)² = 1, which is 4k² + k² = 1, so 5k² = 1, meaning k² = 1/5, so k = ±1/√5. Hmm, but since tanα is negative, that means either sinα is negative and cosα is positive, or sinα is positive and cosα is negative. So depending on the quadrant where alpha is, the signs will change. But maybe I don't need to worry about the signs right now because I'm dealing with squared terms later on.Wait, but in the expression I need to compute, I have sin2α, which is 2sinαcosα. So maybe I can express everything in terms of tanα. Let me recall some trigonometric identities. I know that sin2α is 2tanα/(1 + tan²α), and cos²α is 1/(1 + tan²α). Similarly, sin²α is tan²α/(1 + tan²α). So maybe I can substitute all these into the expression.Let me write down each term:sin2α = 2tanα/(1 + tan²α) = 2*(-2)/(1 + (-2)²) = (-4)/(1 + 4) = (-4)/5.cos²α = 1/(1 + tan²α) = 1/(1 + 4) = 1/5.sin²α = tan²α/(1 + tan²α) = (4)/(5).Now, substituting these into the expression:(sin2α - cos²α)/sin²α = [(-4/5 - 1/5)]/(4/5) = [(-5/5)]/(4/5) = (-1)/(4/5) = (-1)*(5/4) = -5/4.Wait, that seems straightforward. But let me double-check if I did everything correctly.Alternatively, maybe I can express the entire expression in terms of tanα without computing each term separately. Let me try that approach.Starting with the numerator: sin2α - cos²α. I can write sin2α as 2sinαcosα, so the numerator becomes 2sinαcosα - cos²α. Then, factor out cosα: cosα(2sinα - cosα). Hmm, not sure if that helps directly.Alternatively, maybe divide both numerator and denominator by cos²α to express everything in terms of tanα. Let's try that.So, the expression is (sin2α - cos²α)/sin²α. Let me divide numerator and denominator by cos²α:Numerator: (sin2α - cos²α)/cos²α = (2sinαcosα - cos²α)/cos²α = 2sinα/cosα - 1 = 2tanα - 1.Denominator: sin²α/cos²α = tan²α.So, the entire expression becomes (2tanα - 1)/tan²α. Now, substituting tanα = -2:(2*(-2) - 1)/(-2)² = (-4 -1)/4 = (-5)/4 = -5/4.Okay, that's the same result as before. So that confirms it. Therefore, the value is -5/4.I think both methods lead to the same answer, so I feel confident that -5/4 is correct. I don't see any mistakes in my calculations, so I think that's the right answer.**Final Answer**The value is boxed{-dfrac{5}{4}}.

💡First, I need to understand the operation defined by ( a diamond b = sqrt{a^2 + b^2} ). This operation takes two real numbers, squares each of them, adds the squares together, and then takes the square root of the sum.To solve the expression ( (10 diamond 24) diamond ((-24) diamond (-10)) ), I'll break it down into smaller steps.**Step 1: Calculate ( 10 diamond 24 )**[10 diamond 24 = sqrt{10^2 + 24^2} = sqrt{100 + 576} = sqrt{676} = 26]**Step 2: Calculate ( (-24) diamond (-10) )**[(-24) diamond (-10) = sqrt{(-24)^2 + (-10)^2} = sqrt{576 + 100} = sqrt{676} = 26]**Step 3: Apply the operation ( diamond ) to the results from Steps 1 and 2**[(10 diamond 24) diamond ((-24) diamond (-10)) = 26 diamond 26 = sqrt{26^2 + 26^2}]**Step 4: Simplify the expression**[sqrt{26^2 + 26^2} = sqrt{676 + 676} = sqrt{1352}]**Step 5: Recognize that ( sqrt{1352} ) can be rewritten using ( sqrt{2} )**[sqrt{1352} = sqrt{2 times 676} = sqrt{2} times sqrt{676} = 26sqrt{2}]**Step 6: Conclude with the final answer**[26sqrt{2}]

💡Okay, so I have this problem here: If ( g(x) = frac{18}{4 + 2x} ), what is the value of ( left[g^{-1}(3)right]^{-3} )? Hmm, that looks a bit intimidating at first, but I think I can break it down step by step.First off, I need to find the inverse function of ( g(x) ), which is ( g^{-1}(x) ). Once I have that, I can plug in 3 into the inverse function to find ( g^{-1}(3) ). After that, I need to take that result and raise it to the power of -3. So, let's tackle this one step at a time.Starting with finding the inverse function. I remember that to find the inverse of a function, you essentially switch the roles of ( x ) and ( y ) and then solve for ( y ). So, let's set ( y = g(x) ), which gives me:[ y = frac{18}{4 + 2x} ]Now, to find the inverse, I need to solve for ( x ) in terms of ( y ). Let's do that step by step.First, multiply both sides of the equation by ( 4 + 2x ) to get rid of the denominator:[ y(4 + 2x) = 18 ]Expanding the left side:[ 4y + 2xy = 18 ]Now, I want to isolate the terms with ( x ) on one side. So, subtract ( 4y ) from both sides:[ 2xy = 18 - 4y ]Next, solve for ( x ) by dividing both sides by ( 2y ):[ x = frac{18 - 4y}{2y} ]So, that gives me the inverse function ( g^{-1}(y) ):[ g^{-1}(y) = frac{18 - 4y}{2y} ]Alright, now that I have the inverse function, I can plug in 3 for ( y ) to find ( g^{-1}(3) ):[ g^{-1}(3) = frac{18 - 4(3)}{2(3)} ]Calculating the numerator and denominator separately:Numerator: ( 18 - 4(3) = 18 - 12 = 6 )Denominator: ( 2(3) = 6 )So, ( g^{-1}(3) = frac{6}{6} = 1 )Okay, so ( g^{-1}(3) ) is 1. Now, the problem asks for ( left[g^{-1}(3)right]^{-3} ), which is ( 1^{-3} ).Wait a minute, what's ( 1^{-3} )? I remember that a negative exponent means the reciprocal. So, ( 1^{-3} = frac{1}{1^3} ). And since ( 1^3 = 1 ), this simplifies to ( frac{1}{1} = 1 ).So, putting it all together, ( left[g^{-1}(3)right]^{-3} = 1 ).But just to make sure I didn't make any mistakes, let me double-check my steps.1. I started by setting ( y = frac{18}{4 + 2x} ) and solved for ( x ). That gave me ( x = frac{18 - 4y}{2y} ), which seems correct.2. Plugging in 3 for ( y ) gave me ( frac{18 - 12}{6} = 1 ). That checks out.3. Then, raising 1 to the power of -3 is indeed 1, because any number to any power is itself, and the reciprocal of 1 is still 1.So, I think I did everything right. The answer should be 1.**Final Answer**The value of ( left[g^{-1}(3)right]^{-3} ) is boxed{1}.

💡Alright, so I have this problem: "How many 3-digit numbers have the property that the units digit is at least three times the tens digit?" Hmm, okay. Let me try to figure this out step by step.First, let's understand what a 3-digit number is. It's a number from 100 to 999, right? So, it has three places: hundreds, tens, and units. Each of these places can have digits from 0 to 9, but the hundreds place can't be 0 because then it wouldn't be a 3-digit number anymore.Now, the problem is about the relationship between the units digit and the tens digit. Specifically, the units digit needs to be at least three times the tens digit. So, if I denote the tens digit as 'T' and the units digit as 'U', the condition is U ≥ 3T.Okay, so I need to count all 3-digit numbers where this condition holds. Let's break this down.First, let's consider the possible values for the tens digit, T. Since it's a digit, T can be from 0 to 9. However, the hundreds digit, H, can be from 1 to 9 because it can't be 0.For each possible value of T, I need to find how many possible values of U satisfy U ≥ 3T. Then, for each of these combinations, the hundreds digit can be any of the 9 possible digits (1-9). So, the total number of such 3-digit numbers would be the sum over all T of (number of valid U for each T) multiplied by 9.Let me write this down:Total numbers = 9 * (number of valid U for T=0 + number of valid U for T=1 + ... + number of valid U for T=9)So, I need to figure out for each T from 0 to 9, how many U satisfy U ≥ 3T.Let's go through each T:1. **T = 0**: U must be ≥ 0. Since U is a digit from 0 to 9, all 10 values are valid.2. **T = 1**: U must be ≥ 3*1 = 3. So, U can be 3,4,5,6,7,8,9. That's 7 values.3. **T = 2**: U must be ≥ 6. So, U can be 6,7,8,9. That's 4 values.4. **T = 3**: U must be ≥ 9. Since the maximum U is 9, only U=9 is valid. So, 1 value.5. **T = 4**: U must be ≥ 12. But U is a single digit, so U can't be 12 or more. Therefore, no valid U.6. **T = 5**: Similarly, U must be ≥ 15, which is impossible. So, 0.7. **T = 6**: U must be ≥ 18, which is also impossible. 0.8. **T = 7**: U must be ≥ 21, impossible. 0.9. **T = 8**: U must be ≥ 24, impossible. 0.10. **T = 9**: U must be ≥ 27, impossible. 0.So, summarizing the number of valid U for each T:- T=0: 10- T=1: 7- T=2: 4- T=3: 1- T=4 to T=9: 0Now, let's add these up: 10 + 7 + 4 + 1 = 22.So, for each hundreds digit (which has 9 possibilities: 1-9), there are 22 valid combinations of T and U.Therefore, the total number of such 3-digit numbers is 9 * 22 = 198.Wait, let me double-check to make sure I didn't miss anything.- For T=0, U can be 0-9: 10 options.- For T=1, U must be 3-9: 7 options.- For T=2, U must be 6-9: 4 options.- For T=3, U must be 9: 1 option.- For T=4 to T=9, no possible U since 3T exceeds 9.Yes, that seems correct. So, adding those gives 22, and multiplying by 9 for the hundreds digit gives 198.I think that's the answer. But just to be thorough, let me consider if there are any edge cases or if I miscounted somewhere.- When T=0, U can be 0. Is 0 considered a valid digit for U? Yes, because the units digit can be 0. So, that's fine.- When T=1, U starts at 3. So, 3,4,5,6,7,8,9: that's 7 numbers. Correct.- When T=2, U starts at 6: 6,7,8,9: 4 numbers. Correct.- When T=3, U must be 9: only 1 number. Correct.- For T≥4, no possible U since 3*4=12, which is more than 9. So, 0.Yes, everything checks out. So, the total number is indeed 198.