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If n is the product of the integers from 1 to 20 inclusive,

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If n is the product of the integers from 1 to 20 inclusive, [#permalink] New post 22 Nov 2009, 17:05
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If n is the product of the integers from 1 to 20 inclusive, what is the greatest integer k for which \(2^k\) is a factor of n?

A. 10
B. 12
C. 15
D. 18
E. 20
[Reveal] Spoiler: OA
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Re: Number Properties Question [#permalink] New post 22 Nov 2009, 17:13
shanewyatt wrote:
If n is the product of the integers from1 to 20 inclusive, what is the greatest integer k for
which \(2^k\) is a factor of n?
A. 10
B. 12
C. 15
D. 18
E. 20


IMO D

1*2 *3 * 4 * ....20

if we take 2 common out of all even numbers from 1 to 20, we have \(2^{18}\). Hence k=18 because \(2^{18}\)is the max factor in form \(2^K\)
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Re: Number Properties Question [#permalink] New post 22 Nov 2009, 17:25
Expert's post
shanewyatt wrote:
If n is the product of the integers from1 to 20 inclusive, what is the greatest integer k for
which \(2^k\) is a factor of n?
A. 10
B. 12
C. 15
D. 18
E. 20


\(n=20!\) and we need to find greatest integer \(k\), for which \(2^k*n=20!\). Obviously \(k\) would be highest when \(n=1\). So basically we are asked to determine the highest power of \(2\) in \(20!\).

Finding highest power of prime in \(n!\): everything-about-factorials-on-the-gmat-85592.html

Hence the power of \(2\) in \(20!\) would be: \(\frac{20}{2}+\frac{20}{4}+\frac{20}{8}+\frac{20}{16}=10+5+2+1=18\)

Answer: D.
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Re: Number Properties Question [#permalink] New post 22 Nov 2009, 20:01
There are 10 numbers divisible by 2
There are 5 numbers divisible by 4
There are 2 numbers divisible by 8
There is 1 number divisible by 16.

Hence the total number of 2’s in 20! are 10+5+2+1=18
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Re: If n is the product of the integers from 1 to 20 inclusive, [#permalink] New post 19 Apr 2014, 23:49
n= 20!

And

2^K * A = n = 20!

=> 20! = 2^K * A

Mathematically , we need to find the powers of 2 contained in the factorial given.

2/4/6/8/10/12/14/16/18/20

Total contribution of '2's' is 18

Hence the answer
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Re: If n is the product of the integers from 1 to 20 inclusive,   [#permalink] 19 Apr 2014, 23:49
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