help with factorial problems?

Question

1.convert to factorials
2*4*6*8*10
2. what is the largest integer n such that 33! is divisible by 2^n

Answer 1

1. That product can't be expressed as a factorial, but it CAN be expressed as something TIMES a factorial. Here's a hint:

Notice that you can rewrite it as: (2*1)*(2*2)*(2*3)*(2*4)*(2*5). Now rearrange that to (2*2*2*2*2)*(1*2*3*4*5)

2. I can't think of an easy way to do this (maybe somebody else can). The best way I can think of is to actually count the number of 2's that divide into each of the integers from 1 to 33. Like so (the number in parentheses is how many 2's divide into the integer on the left):

1 (0)
2 (1)
3 (0)
4 (2)
5 (0)
6 (1)
7 (0)
8 (3)
9 (0)
10 (1)

...and so on, all the way up to 33. Then add up all the numbers in parentheses, and that's the "n" you want.

Answer 2

Answer to 2. n = 31.

Take a paper and pencil and start solving:

: 33*32*31*30......

:delete all odd numbers so u will be left with
32*30*28*26*.......

:(2^5)*(2*15)*(2^2*7)*(2*13)*........

:take all 2's in one bracket, so u will have:
(2^5*2*2^2*2*2^3*......)

this is nothing but 2^31 i.e. add all the powers

I think I am right, so n shall be 31