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question:When toys are grouped into 12 and 18, always 3 toys remain ungrouped. What is the minimum number of toys to be added so that when they are grouped into 7, no toys will remain ungrouped?

answer:Let's denote the total number of toys as T. According to the problem, when T is divided by 12 and 18, there is a remainder of 3 in both cases. This means that T can be expressed as: T = 12k + 3 = 18m + 3 where k and m are integers. Since 12 = 2^2 * 3 and 18 = 2 * 3^2, the least common multiple (LCM) of 12 and 18 is 2^2 * 3^2 = 36. Therefore, T must be 3 more than a multiple of 36. We can express this as: T = 36n + 3 where n is an integer. Now, we want to find the minimum number of toys to be added to T so that when they are grouped into 7, no toys will remain ungrouped. This means we want to find the smallest non-negative integer x such that: (T + x) mod 7 = 0 Substituting T with 36n + 3, we get: (36n + 3 + x) mod 7 = 0 Since 36 is a multiple of 7 (as 36 mod 7 = 1), we can simplify the equation to: (1n + 3 + x) mod 7 = 0 (3 + x) mod 7 = 0 Now we need to find the smallest non-negative integer x that satisfies this equation. We can do this by testing values of x starting from 0: For x = 0, (3 + 0) mod 7 = 3 mod 7 ≠ 0 For x = 1, (3 + 1) mod 7 = 4 mod 7 ≠ 0 For x = 2, (3 + 2) mod 7 = 5 mod 7 ≠ 0 For x = 3, (3 + 3) mod 7 = 6 mod 7 ≠ 0 For x = 4, (3 + 4) mod 7 = 7 mod 7 = 0 Therefore, the minimum number of toys to be added is boxed{4} . Adding 4 toys to T will result in a total number of toys that is divisible by 7 with no remainder.

question:The complex number z=frac{1+i}{i} (i is the imaginary unit) corresponds to which quadrant in the complex plane? A: First quadrant B: Second quadrant C: Third quadrant D: Fourth quadr

answer:First, let's simplify the given complex number z=frac{1+i}{i}. To do this, multiply both the numerator and the denominator by the conjugate of the denominator, which is -i: z=frac{1+i}{i}timesfrac{-i}{-i}=frac{-i(1+i)}{-itimes i}=frac{-i-i^2}{-i^2}=frac{-i-(-1)}{1}=1-i Now, the complex number z=1-i corresponds to the point (1,-1) in the complex plane. Since both the real part (1) and the imaginary part (-1) are non-zero, we can determine the quadrant by their signs. The real part is positive, and the imaginary part is negative, which indicates that the point (1,-1) lies in the fourth quadrant. Therefore, the answer is boxed{D}. This problem can be solved by understanding the basic arithmetic operations of complex numbers and their geometric representation in the complex plane.

question:Given two different points M(x_1, y_1) and N(x_2, y_2) on the curve of the function y = f(x) with the slopes of the tangents at these points being k_M and k_N respectively, the curvature phi(M, N) between points M and N is defined as phi(M, N) = frac{|k_M k_N|}{|MN|}, where |MN| is the length of the line segment MN. For the curve f(x) = x^3 + 2, find the range of values of phi(M, N) for two different points M(x_1, y_1) and N(x_2, y_2) such that x_1 + x_2 = 2.

answer:For the curve f(x) = x^3 + 2, we have f'(x) = 3x^2. Let x_1 + x_2 = t (|t| > 2). Then, phi(M, N) = frac{3|t|}{sqrt{1 + (t^2 - 1)^2}} = frac{3}{sqrt{t^2 + frac{2}{t^2} - 2}}. Thus, 0 < phi(M, N) < frac{3sqrt{10}}{5}. Therefore, the range of phi(M, N) is boxed{(0, frac{3sqrt{10}}{5})}.

question:The salary of A, B, C, D, and E is Rs. 8000, Rs. 5000, Rs. 14000, Rs. 7000, and Rs. 9000 per month respectively. What is the average salary of A, B, C, D, and E per month?

answer:To find the average salary of A, B, C, D, and E per month, you add up all their salaries and then divide by the number of people. A's salary = Rs. 8000 B's salary = Rs. 5000 C's salary = Rs. 14000 D's salary = Rs. 7000 E's salary = Rs. 9000 Total salary = A + B + C + D + E Total salary = Rs. 8000 + Rs. 5000 + Rs. 14000 + Rs. 7000 + Rs. 9000 Total salary = Rs. 44000 Number of people = 5 Average salary = Total salary / Number of people Average salary = Rs. 44000 / 5 Average salary = Rs. 8800 So, the average salary of A, B, C, D, and E per month is Rs. boxed{8800} .