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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
Posts: 1609
Followers: 6

Kudos [?]: 75 [0], given: 0

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: 1609
Followers: 6

Kudos [?]: 75 [0], given: 0

YeSS!
SVP
Joined: 03 Feb 2003
Posts: 1609
Followers: 6

Kudos [?]: 75 [0], given: 0

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
Posts: 59
Followers: 1

Kudos [?]: 1 [0], given: 0

What's the trick on this one
SVP
Joined: 03 Feb 2003
Posts: 1609
Followers: 6

Kudos [?]: 75 [0], given: 0

Logically 26 is OK, but Excel does not agree. I don't know why.
Manager
Joined: 12 Mar 2003
Posts: 59
Followers: 1

Kudos [?]: 1 [0], given: 0

In Excel I get 62.5. How did you get 26 ?
SVP
Joined: 03 Feb 2003
Posts: 1609
Followers: 6

Kudos [?]: 75 [0], given: 0

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
Joined: 03 Nov 2005
Posts: 397
Location: Chicago, IL
Followers: 3

Kudos [?]: 28 [0], given: 17

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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