💡Okay, so I have this problem here where I need to add two numbers in base 5: (2014)₅ and (223)₅. The result should also be in base 5. Hmm, base 5, that's a little tricky, but I think I can handle it. Let me break it down step by step.First, I remember that adding numbers in a different base is similar to adding in base 10, but I have to be careful with the carries. In base 5, each digit represents powers of 5, so the rightmost digit is 5⁰ (which is 1), then 5¹ (which is 5), then 5² (25), and so on. So, when adding, if the sum of digits in any column reaches 5 or more, I need to carry over to the next higher place value.Let me write down the numbers vertically to align them properly:``` 2 0 1 4+ 2 2 3```Wait, they don't have the same number of digits. To make it easier, I should align them by their least significant digit. So, I'll add leading zeros to the shorter number to match the number of digits. The first number has four digits, and the second has three, so I'll add a zero in front of the second number:``` 2 0 1 4+ 0 2 2 3```Now, both numbers have four digits. Let's add them column by column, starting from the rightmost digit.**Rightmost column (units place):**4 + 3 = 7. But in base 5, 7 is equal to 1 * 5 + 2, so I write down 2 and carry over 1 to the next column.**Next column (fives place):**1 + 2 = 3, plus the carryover of 1 makes it 4. Since 4 is less than 5, I just write down 4 and no carryover.**Next column (twenty-fives place):**0 + 2 = 2. There's no carryover, so I write down 2.**Leftmost column (one hundred twenty-fives place):**2 + 0 = 2. Again, no carryover, so I write down 2.Putting it all together, the sum is 2 2 4 2 in base 5, which is written as (2242)₅.Wait, let me double-check my addition to make sure I didn't make any mistakes. Sometimes, when dealing with different bases, it's easy to mix up the carries.Starting again:``` 2 0 1 4+ 0 2 2 3----------```Adding the rightmost digits: 4 + 3 = 7. In base 5, 7 divided by 5 is 1 with a remainder of 2. So, I write 2 and carry over 1.Next column: 1 + 2 = 3, plus the carryover 1 equals 4. That's correct.Next column: 0 + 2 = 2. No issues there.Leftmost column: 2 + 0 = 2. Still correct.So, the sum is indeed (2242)₅. To be absolutely sure, maybe I can convert both numbers to base 10, add them, and then convert back to base 5 to see if I get the same result.Let's convert (2014)₅ to base 10:Starting from the right:4 * 5⁰ = 4 * 1 = 41 * 5¹ = 1 * 5 = 50 * 5² = 0 * 25 = 02 * 5³ = 2 * 125 = 250Adding them up: 250 + 0 + 5 + 4 = 259.Now, converting (223)₅ to base 10:3 * 5⁰ = 3 * 1 = 32 * 5¹ = 2 * 5 = 102 * 5² = 2 * 25 = 50Adding them up: 50 + 10 + 3 = 63.Adding the two base 10 numbers: 259 + 63 = 322.Now, let's convert 322 back to base 5.To do this, I'll divide 322 by the highest power of 5 less than or equal to 322. The powers of 5 are:5⁰ = 15¹ = 55² = 255³ = 1255⁴ = 625Since 625 is greater than 322, the highest power we'll use is 5³ = 125.322 ÷ 125 = 2 with a remainder of 322 - (2*125) = 322 - 250 = 72.72 ÷ 25 = 2 with a remainder of 72 - (2*25) = 72 - 50 = 22.22 ÷ 5 = 4 with a remainder of 22 - (4*5) = 22 - 20 = 2.2 ÷ 1 = 2 with a remainder of 0.So, writing the coefficients from highest power to lowest, we get 2 (for 125), 2 (for 25), 4 (for 5), and 2 (for 1). Therefore, 322 in base 5 is (2242)₅.This matches the result I got earlier by adding directly in base 5. So, I'm confident that (2014)₅ + (223)₅ equals (2242)₅.Just to recap the steps:1. Align the numbers by their least significant digit, adding leading zeros if necessary.2. Add each column from right to left, carrying over when the sum equals or exceeds the base (5 in this case).3. Verify the result by converting both numbers to base 10, adding them, and converting back to base 5.This method ensures that the addition is accurate and helps catch any mistakes that might have been made during the base 5 addition process.

💡Okay, so I'm trying to figure out this grammar problem here. The sentence is: "Our neighbors gave ______ a baby bird yesterday that hurt ______ when it fell from its nest." And the options are A. us, it; B. us, itself; C. ourselves, itself; D. ourselves, it.Alright, let's break this down. First, I need to understand what's being asked. It looks like it's testing pronoun usage in two different parts of the sentence. The first blank is after "gave," and the second blank is after "hurt."Starting with the first blank: "Our neighbors gave ______ a baby bird." So, the structure here is "gave [something] to [someone]." But in the sentence, it's written as "gave ______ a baby bird." That means the pronoun is the recipient of the action. So, "gave us a baby bird" sounds correct because "us" is the object of the verb "gave." So, "us" would go in the first blank.Now, for the second blank: "that hurt ______ when it fell from its nest." This part is a bit trickier. The subject here is "it," referring to the baby bird. So, the bird hurt something when it fell. But what did it hurt? Itself, right? Because the bird fell from its nest and hurt itself. So, "itself" makes sense here because it's the reflexive pronoun referring back to the bird.So, putting it all together, the first blank should be "us" and the second blank should be "itself." That would make the sentence: "Our neighbors gave us a baby bird yesterday that hurt itself when it fell from its nest."Let me double-check the options. Option B is "us, itself," which matches what I just thought. Option A is "us, it," which would mean the bird hurt "it," but "it" would refer to something else, which doesn't make sense here. Option C is "ourselves, itself," but "ourselves" would imply that the neighbors hurt themselves, which isn't the case. Option D is "ourselves, it," which also doesn't fit because "ourselves" isn't appropriate here.So, I'm pretty confident that the correct answer is B: us, itself.

💡Okay, so I've got this problem here where I need to find the possible values of M given that a and b are rational numbers and ab is not zero. The expression for M is (2|a|)/a + (3b)/|b|. Hmm, let me try to figure this out step by step.First, I remember that the absolute value of a number divided by the number itself gives either 1 or -1, depending on whether the number is positive or negative. So, for example, |a|/a is 1 if a is positive and -1 if a is negative. Similarly, |b|/b is 1 if b is positive and -1 if b is negative.Looking at the expression for M, it's 2 times |a|/a plus 3 times b/|b|. So, I can rewrite M as 2*(|a|/a) + 3*(b/|b|). Since |a|/a and b/|b| can only be 1 or -1, this means that each term in the expression for M can only take on specific values.Let me break it down:1. If a is positive, then |a|/a = 1, so the first term becomes 2*1 = 2.2. If a is negative, then |a|/a = -1, so the first term becomes 2*(-1) = -2.Similarly, for the second term:1. If b is positive, then b/|b| = 1, so the second term becomes 3*1 = 3.2. If b is negative, then b/|b| = -1, so the second term becomes 3*(-1) = -3.Now, I can consider all possible combinations of a being positive or negative and b being positive or negative. There are four possible cases:**Case 1: a is positive and b is positive**- First term: 2- Second term: 3- M = 2 + 3 = 5**Case 2: a is positive and b is negative**- First term: 2- Second term: -3- M = 2 + (-3) = -1**Case 3: a is negative and b is positive**- First term: -2- Second term: 3- M = -2 + 3 = 1**Case 4: a is negative and b is negative**- First term: -2- Second term: -3- M = -2 + (-3) = -5So, from these four cases, the possible values of M are 5, -1, 1, and -5. That means M can be either ±1 or ±5. Looking back at the options given:A: ±5 B: 0 or ±1 C: 0 or ±5 D: ±1 or ±5The correct answer should be D: ±1 or ±5 because that's exactly what we found from the four cases.Wait, just to make sure I didn't miss anything. The problem says a and b are rational numbers, but since we're dealing with absolute values and signs, it doesn't really matter if they're rational or not; the expression only depends on their signs. Also, ab ≠ 0, which just means neither a nor b can be zero, so we don't have to worry about division by zero in the expression. Yeah, I think I covered all the bases here.

💡First, I need to simplify the expression inside the parentheses. I'll add the numbers 4, 4, and 6 together.Next, I'll substitute this sum back into the original expression, replacing the parentheses with the calculated value.Then, I'll perform the division operations separately. I'll divide 14 by 3 and 2 by 3.After obtaining the results of the divisions, I'll subtract the second division result from the first to find the final answer.Finally, I'll present the result in a clear and boxed format.

💡Alright, so I've got this problem here about a rectangle. The sum of two sides is 11, and the sum of three sides is 19.5. I need to find the product of all possible distinct values of the perimeter of such a rectangle. Hmm, okay, let's break this down step by step.First, let me visualize a rectangle. It has two pairs of equal sides, right? So, if I call the lengths of the sides 'a' and 'b', then the rectangle has sides a, b, a, and b. That makes sense. Now, the problem says the sum of two sides is 11. Hmm, does that mean two adjacent sides or two opposite sides? It doesn't specify, so I guess I need to consider both possibilities.Case 1: The sum of two adjacent sides is 11. So, that would be a + b = 11. Then, the sum of three sides is 19.5. Since three sides would be two of one length and one of the other, that would be 2a + b or a + 2b. So, I have two possibilities here as well.Subcase 1a: 2a + b = 19.5Subcase 1b: a + 2b = 19.5Let me start with Subcase 1a: 2a + b = 19.5I have two equations:1. a + b = 112. 2a + b = 19.5I can subtract the first equation from the second to eliminate b:(2a + b) - (a + b) = 19.5 - 11Which simplifies to:a = 8.5Now, substitute a = 8.5 back into the first equation:8.5 + b = 11So, b = 11 - 8.5 = 2.5Okay, so in this case, the sides are 8.5 and 2.5. The perimeter would be 2(a + b) = 2*(8.5 + 2.5) = 2*11 = 22.Now, let's look at Subcase 1b: a + 2b = 19.5Again, starting with:1. a + b = 112. a + 2b = 19.5Subtract the first equation from the second:(a + 2b) - (a + b) = 19.5 - 11Which simplifies to:b = 8.5Substitute b = 8.5 back into the first equation:a + 8.5 = 11So, a = 11 - 8.5 = 2.5Wait, that's the same as before, just swapping a and b. So, the sides are still 8.5 and 2.5, giving the same perimeter of 22. So, in this case, both subcases give the same perimeter.Okay, so that's Case 1 done. Now, let's consider Case 2: The sum of two opposite sides is 11. Since opposite sides are equal, that would mean 2a = 11 or 2b = 11.Subcase 2a: 2a = 11So, a = 5.5Now, the sum of three sides is 19.5. Again, three sides could be 2a + b or a + 2b.Subcase 2a1: 2a + b = 19.5We already know a = 5.5, so:2*5.5 + b = 19.511 + b = 19.5b = 19.5 - 11 = 8.5So, sides are 5.5 and 8.5. Perimeter is 2*(5.5 + 8.5) = 2*14 = 28.Subcase 2a2: a + 2b = 19.5Again, a = 5.5:5.5 + 2b = 19.52b = 19.5 - 5.5 = 14b = 7So, sides are 5.5 and 7. Perimeter is 2*(5.5 + 7) = 2*12.5 = 25.Now, Subcase 2b: 2b = 11So, b = 5.5Again, sum of three sides is 19.5.Subcase 2b1: 2a + b = 19.5b = 5.5:2a + 5.5 = 19.52a = 19.5 - 5.5 = 14a = 7So, sides are 7 and 5.5. Perimeter is 2*(7 + 5.5) = 2*12.5 = 25.Subcase 2b2: a + 2b = 19.5b = 5.5:a + 2*5.5 = 19.5a + 11 = 19.5a = 8.5So, sides are 8.5 and 5.5. Perimeter is 2*(8.5 + 5.5) = 2*14 = 28.Wait a minute, so in Subcases 2a1 and 2b2, we get perimeters of 28 and 25, and in Subcases 2a2 and 2b1, we get perimeters of 25 and 28. So, essentially, the distinct perimeters from Case 2 are 25 and 28.So, putting it all together, from Case 1, we have a perimeter of 22, and from Case 2, we have perimeters of 25 and 28. So, the distinct perimeters are 22, 25, and 28.Now, the problem asks for the product of all possible distinct values of the perimeter. So, I need to multiply these three perimeters together: 22 * 25 * 28.Let me calculate that step by step.First, 22 * 25. Well, 22 * 25 is the same as 22 * (100/4) = (22*100)/4 = 2200/4 = 550.Then, 550 * 28. Let's break that down:550 * 28 = 550 * (20 + 8) = (550 * 20) + (550 * 8)550 * 20 = 11,000550 * 8 = 4,400Adding those together: 11,000 + 4,400 = 15,400.So, the product of all possible distinct perimeters is 15,400.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, 22 * 25:22 * 25 = (20 + 2) * 25 = 20*25 + 2*25 = 500 + 50 = 550. That's correct.Then, 550 * 28:Breaking it down:550 * 28 = 550 * (20 + 8) = 550*20 + 550*8550*20: 550*2=1,100, so 1,100*10=11,000550*8: 550*8=4,400Adding 11,000 + 4,400 gives 15,400. Yep, that's correct.So, I think I've got it right. The distinct perimeters are 22, 25, and 28, and their product is 15,400.

💡Okay, so I have this problem here about a moving point P in the Cartesian coordinate plane. The equation given is (x - 1)² + (y - 3)² = 4. I need to figure out through which quadrants the trajectory of point P passes. The options are A: first and second quadrants, B: second and third quadrants, C: third and fourth quadrants, D: first and fourth quadrants.Alright, let's start by understanding what this equation represents. The general form of a circle in the coordinate plane is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. So, in this case, comparing it to the standard form, the center of the circle should be at (1, 3), and the radius is the square root of 4, which is 2.So, the circle is centered at (1, 3) with a radius of 2. Now, I need to figure out which quadrants this circle passes through. Quadrants in the Cartesian plane are divided based on the signs of the x and y coordinates. The first quadrant is where both x and y are positive, the second quadrant is where x is negative and y is positive, the third quadrant is where both x and y are negative, and the fourth quadrant is where x is positive and y is negative.Since the center of the circle is at (1, 3), which is in the first quadrant, I can already tell that the circle will definitely pass through the first quadrant. But I need to check if it extends into other quadrants as well.The radius is 2, so the circle extends 2 units in all directions from the center. Let's see how far the circle goes in each direction:1. To the left of the center: x-coordinate decreases by 2, so from 1 to 1 - 2 = -1. So, the circle reaches x = -1 on the left side.2. To the right of the center: x-coordinate increases by 2, so from 1 to 1 + 2 = 3. So, the circle reaches x = 3 on the right side.3. Upwards from the center: y-coordinate increases by 2, so from 3 to 3 + 2 = 5. So, the circle reaches y = 5 at the top.4. Downwards from the center: y-coordinate decreases by 2, so from 3 to 3 - 2 = 1. So, the circle reaches y = 1 at the bottom.Now, let's see which quadrants these points fall into:- The leftmost point is at (-1, 3). Here, x is negative and y is positive, so this point is in the second quadrant.- The rightmost point is at (3, 3). Both x and y are positive, so this is still in the first quadrant.- The topmost point is at (1, 5). Again, both x and y are positive, so first quadrant.- The bottommost point is at (1, 1). Both x and y are positive, so first quadrant.So, the circle extends into the second quadrant on the left side but stays within the first quadrant on the right, top, and bottom sides. Therefore, the circle passes through the first and second quadrants.Let me double-check to make sure I didn't miss anything. The center is in the first quadrant, and the radius is 2. Since the x-coordinate of the center is 1, subtracting 2 gives x = -1, which is negative, so it does reach into the second quadrant. The y-coordinate is 3, so subtracting 2 gives y = 1, which is still positive, so it doesn't reach the fourth quadrant. Similarly, adding 2 to y gives y = 5, still positive, so no third quadrant involvement either.Therefore, the circle passes through the first and second quadrants, which corresponds to option A.**Final Answer**boxed{A}