Question

Im sorry for bombarding you guys with questions. I've only needed help with 3 of 40 so far.. Anyways, moving on!

3! = 3 * 2 * 1
4! = 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1
30! = 2^x * N where N is odd.

What is the value of x?

I have no idea where to start, but reading the rest of the question i kind of understand it. The part that I don't understand is what the equation 2^x * N means, and also, how to find the value of x. Thanks for your help.

Answer 1

If you take all the numbers from 1 to 30, there are
15 multiples of 2
7 multiples of 2^2=4
3 multiples of 2^3=8
1 multiple of 2^4=16

So, 2 occurs 15+7+3+1 = 26 times

N = 26

Answer 2

let's look a bit further in the pattern

3! = 3*2*1 -----> = 2^1*3
4! = 4*3*2*1 ---> 8*3 = 2^3 * 3
5! = 5*4*3*2*1 --> 8*15 = 2^3 * 15
6! = 6*5*4*3*2*1 = 16*45 = 2^4 * 45
7! = ...... = 16*315 = 2^4 * 45
8! = ..... we are multiplying by 8 which is 2^3, so we have 2^7 * odd

since we are factoring out powers of 2, the remaining quotient will always be odd

at 8! we have 2^7 * odd
at 9! no new 2 was added
at 10! a new 2 ----> 2^8 * odd
at 12! we get another 2^2 ---> 2^10*odd

I will simply count the increases of the exponent for those that make a change

notice n! does not make a change in the power of 2 if n is odd from the previous factorial

14! --- 1
16! ---- 4
18! ---- 1
20! ---- 2
22! ---- 1
24! ----- 3
26! --- 1
28! ---- 2
30! --- 1
total increases in exponents is 16

so we have
30! = 2^26 * odd