answer:1. **Define the number**: Start with 9,999. Notice that 10,000 = 100^2, hence 9,999 = 10,000 - 1 = 100^2 - 1. 2. **Factorize 9,999**: [ 9,999 = 100^2 - 1^2 = (100 + 1)(100 - 1) = 101 times 99. ] Thus, [ 9,999 = 101 cdot 99. ] 3. **Check for primality and find the greatest prime divisor**: - 101 is a prime number (it is not divisible by any prime numbers less than its square root, which is approximately 10). - 99 = 3^2 times 11, so 99 is composite. 4. **Determine the greatest prime divisor**: The greatest prime divisor of 9,999 is 101. 5. **Calculate the sum of the digits of 101**: [ 1 + 0 + 1 = 2. ] 6. **Conclusion with final answer**: The sum of the digits of the greatest prime number that is a divisor of 9,999 is 2. The final answer is boxed{textbf{(A)} : 2}

answer:To solve this problem, we will use the Chicken McNugget Theorem (also known as the Frobenius Coin Problem). The theorem states that for two coprime integers ( m ) and ( n ), the largest integer that cannot be expressed as ( am + bn ) for nonnegative integers ( a ) and ( b ) is ( mn - m - n ). 1. **Identify the coprime integers:** Here, the integers are 8 and 11, which are coprime (i.e., their greatest common divisor is 1). 2. **Apply the Chicken McNugget Theorem:** According to the theorem, the largest integer that cannot be expressed as ( 8a + 11b ) for nonnegative integers ( a ) and ( b ) is: [ 8 cdot 11 - 8 - 11 = 88 - 19 = 69 ] 3. **Adjust for positive integers:** The problem specifies that ( a ) and ( b ) must be positive integers. To adjust for this, we redefine ( a' = a - 1 ) and ( b' = b - 1 ), where ( a' ) and ( b' ) are nonnegative integers. This means: [ 8a + 11b = 8(a' + 1) + 11(b' + 1) = 8a' + 11b' + 19 ] Therefore, the largest integer ( n ) that cannot be expressed as ( 8a + 11b ) for positive integers ( a ) and ( b ) is: [ n - 19 = 69 implies n = 69 + 19 = 88 ] Conclusion: The largest positive integer ( n ) for which there are no positive integers ( a ) and ( b ) such that ( 8a + 11b = n ) is ( boxed{88} ).

answer:1. **Substitute x = 3 into the equation**: Given that 3 is a root of the polynomial x^3 + hx - 20 = 0, substitute 3 for x: [ 3^3 + h cdot 3 - 20 = 0 ] 2. **Simplify the equation**: Calculate 3^3 and simplify the left-hand side: [ 27 + 3h - 20 = 0 ] [ 7 + 3h = 0 ] 3. **Solve for h**: Isolate h by subtracting 7 from both sides and then dividing by 3: [ 3h = -7 ] [ h = frac{-7}{3} ] 4. **Conclude with the correct answer**: The value of h that makes 3 a root of the polynomial is -frac{7}{3}. Therefore, the correct answer is -frac{7{3}}. The final answer is boxed{-frac{7}{3}}

answer:Solution: The sum of the first n terms of an arithmetic sequence {a_n} is S_n= frac {(a_1+a_n)n}{2}. Since the summation in an arithmetic sequence is analogous to the product in a geometric sequence, Thus, for a geometric sequence {b_n} where all terms are positive, the product of the first n terms T_n=(b_1cdot b_n)^{frac {n}{2}}= sqrt {(b_1cdot b_n)^n}, Therefore, the answer is: boxed{sqrt {(b_1cdot b_n)^n}} By using the formulas for the general term and summation of arithmetic and geometric sequences, and by analogy, we can derive the result. When applying analogy, the summation in an arithmetic sequence is usually analogous to the product in a geometric sequence. This question tests analogy reasoning, and understanding the analogy between arithmetic and geometric sequences. Clarifying the connection and difference between arithmetic and geometric sequences is key to solving this problem.