answer:To solve this problem, compute the number of ways to select two numbers from the set that have a difference of at least 3: - Calculate the total number of ways to choose any two numbers from the set, which is given by binom{9}{2}=36. - Determine the number of pairs with a difference of less than 3. These pairs are those that are either consecutive or separated by only one number. - The pairs with a difference of exactly 1 (consecutive) are (1, 2), (2, 3), ..., (8, 9), totaling 8 pairs. - The pairs with a difference of exactly 2 are (1, 3), (2, 4), ..., (7, 9), totaling 7 pairs. - Sum these to find the number of pairs with differences less than 3: 8 + 7 = 15 pairs. - Subtract from the total number to find those with differences of 3 or more: 36 - 15 = 21 pairs. - Calculate the probability: frac{21}{36} = boxed{frac{7}{12}}.

answer:Since (alpha in (0, frac{pi}{2})) and (cos(alpha + frac{pi}{6}) = frac{4}{5}), then (alpha + frac{pi}{6}) is an acute angle. Therefore, (sin(alpha + frac{pi}{6}) = sqrt{1 - cos^2(alpha + frac{pi}{6})} = frac{3}{5}). Then, (sin(2alpha + frac{pi}{3}) = 2sin(alpha + frac{pi}{6})cos(alpha + frac{pi}{6}) = 2 cdot frac{3}{5} cdot frac{4}{5} = frac{24}{25}). Thus, the correct answer is: (boxed{B}). This problem mainly examines the basic relationships between trigonometric functions of the same angle, the double-angle formula, and the signs of trigonometric functions in different quadrants. It is a basic question.

answer:Let's call the certain number "d". According to the problem, when n is divided by d, the remainder is 1. This can be written as: n = kd + 1 where k is the quotient when n is divided by d. Now, we are told that when 3 times n (which is 3n) is divided by d, the remainder is 3. This can be written as: 3n = jd + 3 where j is the quotient when 3n is divided by d. We can substitute the expression for n from the first equation into the second equation: 3(kd + 1) = jd + 3 Expanding the left side, we get: 3kd + 3 = jd + 3 Since the remainders are equal (both are 3), we can equate the terms that are multiples of d: 3kd = jd Now, we can divide both sides by d (assuming d is not zero): 3k = j This tells us that the quotient j is three times the quotient k. However, we are interested in finding the value of d. Since the remainder when n is divided by d is 1, and the remainder when 3n is divided by d is 3, we can infer that d must be greater than 3 (otherwise, the remainder could not be 3). Also, since the remainder when n is divided by d is 1, multiplying n by 3 would multiply the remainder by 3 as well, as long as 3 times the remainder is less than d. Therefore, the number d must be such that 3 times the remainder when n is divided by d (which is 1) is less than d. The smallest number that satisfies this condition is 4, because 3 times 1 is 3, which is less than 4. So, the number d is boxed{4} .

answer:Analysis of the problem: All natural numbers are integers, and 4 is a natural number, so 4 is an integer. Major premise: The statement that all natural numbers are integers is correct, Minor premise: The statement that 4 is a natural number is also correct, Conclusion: The statement that 4 is an integer is correct, ∴ This reasoning is correct, hence the choice is A. Key point: Conducting simple deductive reasoning. boxed{text{A}}