answer:1. **Define the variables**: Let the two whole numbers be x and y. We know one of these two numbers is even. 2. **Set up the equations**: - The product of the numbers is 40: ( xy = 40 ) - The sum of the numbers is 13: ( x + y = 13 ) 3. **Express one variable in terms of the other**: - From the sum equation, solve for x: ( x = 13 - y ) 4. **Substitute the expression for x into the product equation**: [ (13 - y)y = 40 ] 5. **Simplify and form a quadratic equation**: [ 13y - y^2 = 40 implies y^2 - 13y + 40 = 0 ] 6. **Factorize the quadratic equation**: [ (y - 5)(y - 8) = 0 ] 7. **Solve for y**: - Setting each factor equal to zero gives: [ y - 5 = 0 quad text{or} quad y - 8 = 0 ] - Thus, ( y = 5 ) or ( y = 8 ). 8. **Find the corresponding values of x**: - If ( y = 5 ), then ( x = 13 - 5 = 8 ) (which is even, satisfying the condition). - If ( y = 8 ), then ( x = 13 - 8 = 5 ) (but y is even, so this also meets the condition). 9. **Identify the larger number**: - The possible values for the larger number are 8 and 5. The larger number is therefore 8. 10. **Conclude with the answer**: - The larger number is ( 8 ). The final answer is boxed{8} (Choice B).

answer:Writing out the recursive statement for b_n, b_{n-1}, dots, b_{15} and summing them gives [b_n + dots + b_{15} = 150(b_{n-1} + dots + b_{15}) + n + dots + 15] which simplifies to [b_n = 149(b_{n-1} + dots + b_{15}) + frac{1}{2}(n+15)(n-14)] Therefore, b_n is divisible by 143 if and only if frac{1}{2}(n+15)(n-14) is divisible by 143, so (n+15)(n-14) needs to be divisible by 11 and 13 (since 143 = 11 * 13). Assume that n+15 is a multiple of 13. Writing out a few terms, n=16, 29, 42, 55, we see that n=55 is the smallest n that works in this case. Next, assume that n-14 is a multiple of 13. Writing out a few terms, n=27, 40, 53, 66, we see that n=27 is the smallest n that works in this case. The smallest n is boxed{27}. Note that we can also construct the solution using CRT by assuming either 13 divides n+15 and 11 divides n-14, or 11 divides n+15 and 13 divides n-14, and taking the smaller solution.

answer:Solution: The function f(x) = cos x sin x = frac{1}{2}sin 2x, Since it is an odd function and also a periodic function, proposition 1 is incorrect; The period of the function is pi, so proposition 2 is incorrect; Proposition 3 is correct as f(x) is an increasing function in the interval left[-frac{pi}{4}, frac{pi}{4}right]; Proposition 4 is correct because the graph of f(x) is symmetric about the line x = frac{3pi}{4}. When x = frac{3pi}{4}, f(x) reaches its minimum value, which is the axis of symmetry. Therefore, the answer is: boxed{text{Propositions 3 and 4}} Simplifying the function f(x) = cos x sin x to f(x) = frac{1}{2}sin 2x, we can judge the correctness of proposition 1 using the property of odd functions; judge the correctness of proposition 2 using the period of the function; judge proposition 3 using monotonicity, and judge proposition 4 using symmetry. This question is a basic one, examining the simplification of trigonometric functions and the properties of basic functions. Mastering the properties of basic functions is the basis for solving this problem, and strengthening the learning of basic knowledge is essential for improving the application of mathematical knowledge.

answer:Given f(x) = (x+3)(x+2)(x+1)x(x-1)(x-2)(x-3), thus f'(x) = (x+3)(x+2)(x+1)x(x-2)(x-3) + (x-1)[(x+3)(x+2)(x+1)x(x-2)(x-3)]', therefore f'(1) = (1+3)(1+2)(1+1)(1-2)(1-3) = 48, hence, the answer is: boxed{B}. By the rules of differentiation, f'(x) = (x+3)(x+2)(x+1)x(x-2)(x-3) + (x-1)[(x+3)(x+2)(x+1)x(x-2)(x-3)]', substituting x=1 into it directly gives the value of f'(1). This question tests the calculation, the rules of differentiation, and computational skills, and is considered a basic question.