💡To determine the smallest six-digit number composed of the digits 1, 2, 3, 4, 5, and 6 that is divisible by 6, I need to consider the criteria for divisibility by 6. A number is divisible by 6 if it is divisible by both 2 and 3.First, I check for divisibility by 3. The sum of the digits is 1 + 2 + 3 + 4 + 5 + 6 = 21, which is divisible by 3. Therefore, any permutation of these digits will be divisible by 3.Next, I check for divisibility by 2. A number is divisible by 2 if its last digit is even. The even digits available are 2, 4, and 6. To find the smallest number, I want the smallest possible digits to be in the higher place values. I start by placing the smallest even digit, 2, at the end. This gives me the number 123452. However, since all digits must be unique, this is not valid.Next, I try placing the next smallest even digit, 4, at the end. This results in the number 123454, which also repeats the digit 4, so it's not valid.Finally, I place the largest even digit, 6, at the end. This gives me the number 123456. All digits are unique, and the last digit is even, satisfying both divisibility rules.Therefore, the smallest six-digit number using each digit 1, 2, 3, 4, 5, and 6 exactly once that is divisible by 6 is 123456.

💡First, calculate the total number of pages Robert read during each segment of the ten-day period.For the first three days:[3 text{ days} times 28 text{ pages/day} = 84 text{ pages}]For the next three days:[3 text{ days} times 35 text{ pages/day} = 105 text{ pages}]For the following three days:[3 text{ days} times 42 text{ pages/day} = 126 text{ pages}]On the last day:[15 text{ pages}]Now, sum all the pages read over the ten days:[84 + 105 + 126 + 15 = 330 text{ pages}]Therefore, the total number of pages in the book is (boxed{330}).

💡First, I need to determine the initial amount of money before spending. Let's denote this amount as ( X ).You spent 30% of ( X ), which means you have 70% of ( X ) remaining because ( 100% - 30% = 70% ).The remaining 70% of ( X ) is equal to 2800. This can be written as:[0.70 times X = 2800]To find ( X ), I will divide 2800 by 0.70:[X = frac{2800}{0.70} = 4000]Therefore, the initial amount of money was 4000.

💡Okay, so I've got this problem here: Lee can make 18 cookies with two cups of flour. How many cookies can he make with three cups of flour? Hmm, let's see how to approach this.First, I need to understand the relationship between the amount of flour and the number of cookies. It seems like more flour should mean more cookies, right? So, if two cups make 18 cookies, three cups should make more than 18. But how much more?Maybe I can set up a proportion to figure this out. Proportions are useful when dealing with ratios, and in this case, the ratio of cookies to cups of flour should stay the same. So, if 18 cookies correspond to 2 cups, then x cookies should correspond to 3 cups.Let me write that out:18 cookies / 2 cups = x cookies / 3 cupsSo, 18/2 = x/3Now, to solve for x, I can cross-multiply. That means 18 times 3 equals 2 times x.18 * 3 = 2 * xCalculating 18 times 3... 18 times 3 is 54.So, 54 = 2xNow, to find x, I need to divide both sides by 2.54 / 2 = x54 divided by 2 is 27.So, x = 27Therefore, Lee can make 27 cookies with three cups of flour.Wait, let me double-check that. If two cups make 18 cookies, then one cup makes 9 cookies because 18 divided by 2 is 9. So, three cups would make 9 times 3, which is 27. Yeah, that makes sense.I think that's the right answer. It's always good to check my work to make sure I didn't make any mistakes. So, both methods—using proportions and calculating per cup—give me the same result. That gives me confidence that 27 is the correct number of cookies Lee can make with three cups of flour.

💡To find the imaginary part of the complex number ( z ) that satisfies the equation ( (1 - i)z = 3 + 7i ), we can follow these steps:1. **Express ( z ) in terms of the given equation:** [ z = frac{3 + 7i}{1 - i} ]2. **Multiply the numerator and the denominator by the conjugate of the denominator to simplify:** [ z = frac{(3 + 7i)(1 + i)}{(1 - i)(1 + i)} ]3. **Expand the numerator and the denominator:** [ text{Numerator: } (3 + 7i)(1 + i) = 3(1) + 3(i) + 7i(1) + 7i(i) = 3 + 3i + 7i + 7i^2 ] [ text{Denominator: } (1 - i)(1 + i) = 1^2 - (i)^2 = 1 - (-1) = 2 ]4. **Simplify the numerator using ( i^2 = -1 ):** [ 3 + 3i + 7i + 7(-1) = 3 + 10i - 7 = -4 + 10i ]5. **Divide the simplified numerator by the denominator:** [ z = frac{-4 + 10i}{2} = -2 + 5i ]6. **Identify the imaginary part of ( z ):** The imaginary part of ( z ) is ( 5 ).

💡Okay, so I have this problem here where I need to find the sum of x, y, and z. The problem gives me three logarithmic equations, each set equal to zero. Let me write them down to make it clearer:1. log₃(log₄(log₅ x)) = 02. log₄(log₅(log₃ y)) = 03. log₅(log₃(log₄ z)) = 0Hmm, logarithms inside logarithms. That seems a bit complicated, but maybe I can break it down step by step. I remember that if log_b(a) = 0, then a must be 1 because b^0 = 1. So, maybe I can use that property to simplify each equation.Starting with the first equation: log₃(log₄(log₅ x)) = 0. Since the log base 3 of something is 0, that something must be 1. So, log₄(log₅ x) = 1. Now, applying the same logic, since log base 4 of something is 1, that something must be 4^1 = 4. So, log₅ x = 4. That means x is 5^4. Calculating that, 5^4 is 5*5*5*5, which is 25*25, so 625. Okay, so x is 625.Moving on to the second equation: log₄(log₅(log₃ y)) = 0. Again, since log base 4 of something is 0, that something is 1. So, log₅(log₃ y) = 1. Now, log base 5 of something is 1, so that something is 5^1 = 5. Therefore, log₃ y = 5. That means y is 3^5. Calculating that, 3^5 is 3*3*3*3*3, which is 9*9*3, so 81*3, which is 243. So, y is 243.Now, the third equation: log₅(log₃(log₄ z)) = 0. Log base 5 of something is 0, so that something is 1. Therefore, log₃(log₄ z) = 1. Log base 3 of something is 1, so that something is 3^1 = 3. Thus, log₄ z = 3. So, z is 4^3. Calculating that, 4^3 is 4*4*4, which is 16*4, so 64. Therefore, z is 64.Now, I need to find the sum x + y + z. Plugging in the values I found:x = 625y = 243z = 64Adding them together: 625 + 243 is 868, and 868 + 64 is 932. So, the sum x + y + z is 932.Let me double-check my calculations to make sure I didn't make any mistakes. For x, log₃(log₄(log₅ x)) = 0. So, log₄(log₅ x) = 1, which means log₅ x = 4, so x = 5^4 = 625. That seems right.For y, log₄(log₅(log₃ y)) = 0. So, log₅(log₃ y) = 1, which means log₃ y = 5, so y = 3^5 = 243. That also looks correct.For z, log₅(log₃(log₄ z)) = 0. So, log₃(log₄ z) = 1, which means log₄ z = 3, so z = 4^3 = 64. Yep, that's correct too.Adding them up again: 625 + 243 is 868, plus 64 is indeed 932. So, I think I did everything right. I don't see any mistakes in my steps.Just to make sure, let me think about the properties of logarithms again. If log_b(a) = c, then a = b^c. So, starting from the outermost logarithm, each time I set the inside equal to the base raised to the power of the logarithm's result. Since all the equations are set to 0, each time the inside becomes 1, then the next layer becomes the base raised to 1, and so on. It seems consistent.I guess another way to look at it is to work from the outside in. For example, in the first equation, log₃(something) = 0 implies something = 1. Then, log₄(something else) = 1 implies something else = 4, and then log₅ x = 4 implies x = 5^4. The same logic applies to the other two equations. So, yeah, I think my approach is solid.I also remember that sometimes when dealing with nested logarithms, it's easy to mix up the order or the bases, but in this case, I carefully followed each step, making sure to apply the logarithm properties correctly each time. So, I feel confident that my answer is correct.Just to recap:1. log₃(log₄(log₅ x)) = 0 ⇒ log₄(log₅ x) = 1 ⇒ log₅ x = 4 ⇒ x = 5^4 = 6252. log₄(log₅(log₃ y)) = 0 ⇒ log₅(log₃ y) = 1 ⇒ log₃ y = 5 ⇒ y = 3^5 = 2433. log₅(log₃(log₄ z)) = 0 ⇒ log₃(log₄ z) = 1 ⇒ log₄ z = 3 ⇒ z = 4^3 = 64Sum: 625 + 243 + 64 = 932Everything checks out. I don't think I made any calculation errors, and the logic follows correctly. So, I'm pretty sure the answer is 932.