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Factor of 2^k with n!

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Factor of 2^k with n!

by cr1985 » Wed Jun 01, 2011 11:46 am
Hey guys, I'm new here, but I hope to post often. I have come in contact with many sets of high level questions, so I hope to possibly expose some of these with you all as I work through them. This one, however, should only be medium level, yet I still don't understand how to approach it.

Looking through several posts on factorials, I've realized that they usually do not ask for calculations any bigger than 7!, so I've come to the conclusion that the following question must require some alternate method. Hoping someone can help. Thanks.

If n is the product of the integers from 1 to 20 inclusive, what is the greatest integer k for which 2k is a factor of n?

A. 10; B. 12; C. 15; D: 18; E: 20
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by SoCan » Wed Jun 01, 2011 2:35 pm
The title says 2^k, but the question says 2k, which threw me off for a bit because I didn't notice the title.

You want to know how many multiples of 2 there are in 20! You can ignore the odd numbers, obviously.

Going from 2 to 20, you get

2 2^2 2 2^3 2 2^2 2 2^4 2 2^2

Add the multiples of 2 together and you get 18, so D.

Also, I recommend posting questions in the PS or DS subforums - they're more likely to get looked at quickly over there.
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by cr1985 » Wed Jun 01, 2011 3:12 pm
Awesome. Yes, I meant 2^k, sorry. Now I understand, but I'm just a little confused about the number 18. I'm assuming that the multiples added up plus 0 is what makes it 18 rather than 17, right? And, thanks for the tip on where to post. I will surely be posting some new questions there soon. Thanks.
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by SoCan » Wed Jun 01, 2011 3:47 pm
cr1985 wrote:Awesome. Yes, I meant 2^k, sorry. Now I understand, but I'm just a little confused about the number 18. I'm assuming that the multiples added up plus 0 is what makes it 18 rather than 17, right? And, thanks for the tip on where to post. I will surely be posting some new questions there soon. Thanks.
I listed 18 above, although the formatting may make it tough to read. This is more clear - listing the even numbers with the number of factors of 2:

2(1), 4(2), 6(1), 8(3), 10(1), 12(2), 14(1), 16(4), 18(1), 20(2)

1+2+1+3+1+2+1+4+1+2 = 18
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by cr1985 » Wed Jun 01, 2011 6:42 pm
Ah yes, I was able to tell what you had written in the prior response, but for some reason was counting up to 17 the first few times. I must of missed one. Got it though, thanks.
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by SoCan » Wed Jun 01, 2011 8:17 pm
cr1985 wrote:Ah yes, I was able to tell what you had written in the prior response, but for some reason was counting up to 17 the first few times. I must of missed one. Got it though, thanks.
No problem - when you said 17, I counted again to make sure I hadn't left anything out, and I even counted 17 the first time.
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