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# 3^X is the greatest positive divisor of 10!*20!*30!. Find X

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SVP
Joined: 03 Feb 2003
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3^X is the greatest positive divisor of 10!*20!*30!. Find X [#permalink]  05 May 2003, 05:16
3^X is the greatest positive divisor of 10!*20!*30!. Find X.
SVP
Joined: 03 Feb 2003
Posts: 1607
Followers: 6

Kudos [?]: 84 [0], given: 0

[#permalink]  05 May 2003, 05:24
YeSS!
SVP
Joined: 03 Feb 2003
Posts: 1607
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Kudos [?]: 84 [0], given: 0

[#permalink]  20 May 2003, 02:27
I think there is a mistake here. I have decided to use Excel to check the problem and got that 3^90 is the largest divisor.

WHY?
Manager
Joined: 12 Mar 2003
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[#permalink]  20 May 2003, 04:14
What's the trick on this one
SVP
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[#permalink]  20 May 2003, 04:16
Logically 26 is OK, but Excel does not agree. I don't know why.
Manager
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[#permalink]  20 May 2003, 04:20
In Excel I get 62.5. How did you get 26 ?
SVP
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[#permalink]  20 May 2003, 04:40
10!=1*2*3*4*5*6*7*8*9*10

multiples of 3 are: 3, 6=2*3, 9=3*3. so there are 4 threes, or 3^4.

20! includes the above 3^4, plus threes in 12, 15, 18, other 3^4. Total 3^8

30! includes the above 3^8, plus threes in 21, 24, 27(3^3), 30. Total 3^14.

Overall, 3^(4+8+14)=3^26 -- the largest divisor.

Seems OK, but Excel does not agree
Senior Manager
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[#permalink]  16 Dec 2005, 21:06
x=4+8+14=26

1) 10/3=3 2)20/3=6 3)30/3=10
3/3=1 6/3=2 10/3=3
1/3=0 2/3=0 3/3=1
1/3=0
_________________

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[#permalink] 16 Dec 2005, 21:06
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# 3^X is the greatest positive divisor of 10!*20!*30!. Find X

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