answer:To find the number of odd divisors of ( n ), we need to consider only the prime factors ( p_1, p_2, ) and ( p_3 ) because ( 2^k ) will only contribute to even divisors. The number of divisors of a number ( n ) can be found by adding 1 to each of the exponents in its prime factorization and then multiplying these numbers together. For example, if ( n = p^a cdot q^b ), where ( p ) and ( q ) are distinct primes, then the number of divisors of ( n ) is ( (a+1)(b+1) ). Given that ( n = 2^k cdot p_1^a cdot p_2^b cdot p_3^c ), the number of odd divisors would be the number of divisors of ( p_1^a cdot p_2^b cdot p_3^c ). This is because the factor of ( 2^k ) only contributes even divisors, and we are not interested in those for this question. So, the number of odd divisors of ( n ) is ( (a+1)(b+1)(c+1) ). However, we have the constraints ( a + b > c ) and ( 1 leq b leq c ). These constraints do not change the formula for the number of odd divisors, but they do limit the possible values of ( a, b, ) and ( c ). Since ( a, b, ) and ( c ) are positive integers, the smallest value for each is 1. The constraints do not provide a maximum value for ( a ) or ( c ), but they do imply that ( b ) cannot be greater than ( c ). Therefore, the number of odd divisors of ( n ) is still ( boxed{(a+1)(b+1)(c+1)} ), but with the understanding that ( a, b, ) and ( c ) must satisfy the given constraints.

answer:According to the problem, student A has 3 choices for colleges, student B has 2 choices (since B must choose a different college from A), and the third student has 3 choices. By applying the principle of counting steps, the total number of application outcomes is 3 times 2 times 3 = 18. Therefore, the correct answer is boxed{A}.

answer:Since a>0, b>0, when 0<aleq b, then 3^a<3^b, and 2a<3b. Therefore, 3^a+2a<3^b+3b, which means only option B is correct. Thus, the answer is: boxed{B}. **Analysis:** Given a>0, b>0, when 0<aleq b, then 3^a<3^b, and 2a<3b. It can be derived that 3^a+2a<3^b+3b, which leads to the conclusion.

answer:1. **Rewrite the inequality**: Start by rewriting the given inequality: [ 3x^2 + 7x < 6 ] Subtract 6 from both sides to set the inequality to zero: [ 3x^2 + 7x - 6 < 0 ] 2. **Factor the quadratic expression**: Next, factor the quadratic expression: [ 3x^2 + 7x - 6 = (3x - 2)(x + 3) ] We find the factors by looking for two numbers that multiply to 3 times (-6) = -18 and add to 7. The numbers 9 and -2 work, so we split the middle term: [ 3x^2 + 9x - 2x - 6 = 3x(x + 3) - 2(x + 3) = (3x - 2)(x + 3) ] 3. **Analyze the critical points**: The critical points where the expression changes sign are the roots of the equation (3x - 2)(x + 3) = 0. Solving for x gives: [ 3x - 2 = 0 quad Rightarrow quad x = frac{2}{3} ] [ x + 3 = 0 quad Rightarrow quad x = -3 ] 4. **Determine the sign of the product in each interval**: We test the sign of the product (3x - 2)(x + 3) in the intervals determined by the roots x = -3 and x = frac{2}{3}: - For x < -3, choose x = -4: (3(-4) - 2)((-4) + 3) = (-14)(-1) = 14 (positive). - For -3 < x < frac{2}{3}, choose x = 0: (3(0) - 2)(0 + 3) = (-2)(3) = -6 (negative). - For x > frac{2}{3}, choose x = 1: (3(1) - 2)(1 + 3) = (1)(4) = 4 (positive). 5. **Conclude the solution**: The product (3x - 2)(x + 3) is negative (and thus satisfies the inequality) in the interval where -3 < x < frac{2}{3}. Thus, the solution to the inequality 3x^2 + 7x < 6 is: [ -3 < x < frac{2{3}} ] The final answer is boxed{C}