Appearance
question:An integer n between 1 and 99, inclusive, is to be chosen at random. What is the probability that n(n + 1) will be divisible by 2?
answer:For the product n(n + 1) to be divisible by 2, at least one of the numbers n or n + 1 must be even, because the product of an even number with any other integer is always even. Since n is an integer between 1 and 99, n + 1 will be an integer between 2 and 100. In any pair of consecutive integers, one is always even and the other is odd. Therefore, for any value of n, either n or n + 1 will be even. This means that for every integer n chosen between 1 and 99, the product n(n + 1) will always be divisible by 2. Hence, the probability that n(n + 1) will be divisible by 2 is boxed{1,} or 100%.
question:Let T be the set of 8tuples (b_0, b_1, ..., b_7), where each entry is 0 or 1. Thus, T contains 2^8 8tuples. For each 8tuple t = (b_0, b_1, ..., b_7) in T, let q_t(x) be the polynomial of degree at most 7 such that [q_t(n) = b_n] for 0 le n le 7. Find [sum_{t in T} q_t(8).]
answer:Let [q(x) = sum_{t in T} q_t(x).] For any n, 0 le n le 7, [q(n) = sum_{t in T} q_t(n) = 2^7 = 128,] because q_t(n) = 0 for 128 polynomials q_t(x), and q_t(n) = 1 for 128 polynomials q_t(x). Thus, q(x) = 128, for 8 different values n = 0, 1, 2, ..., 7. Since q(x) must be a polynomial of degree at most 7 and it takes the same value for 8 distinct points, by the Identity Theorem, q(x) = 128 for all x. In particular, q(8) = boxed{128}.
question:Split 99 into four numbers so that the first number plus 2, the second number minus 2, the third number times 2, and the fourth number divided by 2, all yield the same result. What are these four numbers?
answer:Let the equal result be x. Therefore, the other numbers are (x-2), (x+2), frac{x}{2}, and 2x. Thus, we have (x-2) + (x+2) + frac{x}{2} + 2x = 99. Solving this equation, we find x = 22. Therefore, x-2 = 20, x+2 = 24, frac{x}{2} = 11, and 2x = 44. Hence, the answer is boxed{20, 24, 11, 44}.
question:The sum of the lengths of all the edges of a cube is 48 cm. Find the number of cubic centimeters in the volume of the cube.
answer:1. Calculate the length of one edge: A cube has 12 edges, each of equal length. Therefore, if the total length of all edges is 48 cm, then one edge is: [ text{Edge length} = frac{48 text{ cm}}{12} = 4 text{ cm} ] 2. Calculate the volume of the cube: The volume (V) of a cube is given by the cube of the length of its side: [ V = text{Edge length}^3 = 4^3 = 4 times 4 times 4 = 64 text{ cm}^3 ] Conclusion: The volume of the cube is boxed{64 text{ cm}^3}.