Bunuel wrote:
For any integer k greater than 1, the symbol k* denotes the product of all integers between 1 and k, inclusive. If k* is a multiple of 3,675, what is the least possible value of k?
(A) 12
(B) 14
(C) 15
(D) 21
(E) 25
Are You Up For the Challenge: 700 Level Questions: 700 Level QuestionsK* denotes ---> K!
Concept: for K! to be a Multiple of 3,675 ---> K! must be divisible by Each of the Factors of 3,675
in other words, if K! is divisible by 3,675's Prime Factorization, then K! will be divisible by Each of the Factors of 3,675 ----> and will thus be Divisible by 3,675 itself
3,675 = 25 * 147 = (5)(5) * (7) (21) = (3) * (5)^2 * (7)^2
K! is the Multiplication of: (1) (2) (3) (4) ...... up through *(K)
what is the minimum Factorial that will include 2 Prime Factors of 7 in its Prime Factorization?
(*note* if the Factorial includes two 7's in its Prime Factorization, it will surely contain two 5's and one 3)
we need at least 2 Multiples of 7 in the Multiplication
14! = 1 * 2 * 3 .... * 7 ..... * 14
since 7 and 14 are the first two multiples of 7 ----> 14! is the minimum value that is divisible by 3,675