SOLUTION: The prime factorization of "k!" is {{{ 2^25 }}} x {{{ 3^13 }}} x {{{ 5^6 }}} x {{{ 7^4 }}} x {{{ 11^2 }}} x {{{ 13^2 }}} x 17 x 19 x 23. The value for "k" is

Looking at the factors (multipliers)    and  17, we can conclude that  26<= k < 34.


Looking at the factor ,  we can exacerbate this inequality  26 <= k < 33.


Looking at the factor ,  we can exacerbate this inequality  28 <= k < 33.


Looking at the factor ,  we can exacerbate this inequality  28 <= k < 30.


Looking at the factor ,  we can confirm this inequality  28 <= k <= 29.


Since the prime number 29 is not presented in the k! factorization, we conclude that 28 <= k < 29,

             which means that k = 28.


CHECK.  Check it on your own that k = 28 provides the degree of 2 equal to 24 in the factorization of k!


ANSWER.  (a) The condition has a  stating that the index of 2 is 25: it is 24.


         (b)  With the corrected prime factorization of k! = ,  the value of k is 28.