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# If p is the product of the integers from 1 to 30, inclusive,

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Senior Manager
Joined: 18 Jun 2007
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If p is the product of the integers from 1 to 30, inclusive, [#permalink]  28 Jun 2007, 12:04
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18
[/quote]
Director
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Re: root problem [#permalink]  28 Jun 2007, 12:10
rishi2377 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

is 3k = 3^k? if so, k = 14.

C.
Senior Manager
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Quote:
rishi2377 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

is 3k = 3^k? if so, k = 14.

C.

Himalayan, let's assume 3k = 3^k. Can you explain how in the world you figured out it was 14????
Manager
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the number i got after multiplying 1 to 31:
8,222,838,654,177,920,000,000,000,000,000,000.00

so if we divide this number for:

3k if 3k is:

30 274,094,621,805,931,000,000,000,000,000,000.00
36 228,412,184,838,276,000,000,000,000,000,000.00
42 195,781,872,718,522,000,000,000,000,000,000.00
48 171,309,138,628,707,000,000,000,000,000,000.00
54 152,274,789,892,184,000,000,000,000,000,000.00

NOW , HOW CAN I DO THIS WITHOUT COMPUTER I DON'T KNOW
Manager
Joined: 23 Apr 2007
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Re: root problem [#permalink]  28 Jun 2007, 13:08
rishi2377 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18
[/quote]

I go in for 'E'

P=30! (yes 30 factorial)
taking the given choices 3*18 is a factor of 30! and since 18 is the highest choice given thats the ans.
Senior Manager
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Agree with the above.

3*18 is definitely a factor of 30!
Current Student
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well..if the questions asks 3k..then K=18 would work ...

however, if the question asks 3^k, which is more interesting..then.. here is what we do..

question says if p=30!, and 3^k is a factor, what is the largest value of K that we can have..in other words whats the highest power of 3....if you are smart I already answered the question...if you are like me..here is what we do:

30/3 =10 3s...

but we have an additional 3 in 3^2; i.e 9, so 30/9=3 more 3s

and we have one more 3 in 3^3; i.e 27...=1 more 3...

so in all we have 10+3+1 =14 3s....

Senior Manager
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hello folks,

It was actually 3^k. Sorry, I should be more clear.this was my first post on this site and I will be more cautious in the future.

Thx for the help.
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