Preamble: A lot of integer property questions can be solved using prime factorization.saadiagha wrote:If Z is an integer and Z! is divisible by 340, what is the smallest possible value of z?
Hey guys, was stuck on this question, I thought I got the answer correct but it didint seem to match the answer given in the book.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N
Examples:
24 is divisible by 3 <--> 24 = 2x2x2x3
70 is divisible by 5 <--> 70 = 2x5x7
330 is divisible by 6 <--> 330 = 2x3x5x11
56 is divisible by 8 <--> 56 = 2x2x2x7
So, for Z! to be divisible by 340, it must be the case that 340 is hiding in the prime factorization of Z!
Since 340 = (2)(2)(5)(17), we can see that the prime factorization of Z! must include a 17 (and two 2's and a 5).
This tells us that the smallest possible value of Z is 17.
Cheers,
Brent
