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

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Senior Manager
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If p is the product of the integers from 1 to 30, inclusive, [#permalink]  15 Jan 2006, 10:00
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If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18
Director
Joined: 17 Dec 2005
Posts: 548
Location: Germany
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Kudos [?]: 15 [0], given: 0

For questions like this, proceed as follows:

3^k should be a factor of p. Rephrased: How many factors of 3 (that's what 3^k is meant to be) are in p?

Find all multiples of three in p.

3,6,9,12,15,18,21,24,27,30

Now factor out

3,2*3,3^2 etc.

You will see that 3 is fourteen times in p

k=14

This is pretty fast if you're conscious what you're doing.
SVP
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C. 14.

VP
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Re: PS factors and exponents. [#permalink]  15 Jan 2006, 11:01
C. 14

the number of 3 in P is the value of k.
3 = 1x3
6 = 2x3
9 = 3x3
12 = 4x3
15= 5x3
18= 2x3x3
21=7x3
24=8x3
27=3x3x3
30=10x3

altogather fourteen (14) 3's are included in p. so k = 14.
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Number - # of power of 3
3 - 1
6 - 1
9 - 2
12 - 1
15 - 1
18 - 2
21 - 1
24 - 1
27 - 3
30 - 1
Total = 14
Senior Manager
Joined: 05 Jan 2006
Posts: 382
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Got 14 with same approach as others! Seems intuitive and easy!

Though I was thinking can there be another approach! I can not use such simple calculation for such questions for example if it's 1:100 and 5^k than what!

So was thinking as follow
in 1: 30.... 10 No divided by 3
in 1: 30, ...9,18,27......3 No divided by 3*3
in 1:30....27....1 No divided by 3*3*3

total 10+3+1 = 14

This works for 1:100 divided by 5
1:100.... 20 divided by 5
1:100.... 4 divided by 5*5

total 24 !

Do you see any major flow in this method! (Except, I myself prefer first method over this, just because itâ€™s intuitive, and I can not derive such method in exam..
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Re: PS factors and exponents. [#permalink]  17 Jan 2006, 22:44
sperumba wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

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

!30/3^k
k=30/3=10
10/3=3
3/3=1==>10+3+1=14
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Posts: 548
Location: Germany
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Re: PS factors and exponents. [#permalink]  18 Jan 2006, 01:35
rlevochkin wrote:
sperumba wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

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

!30/3^k
k=30/3=10
10/3=3
3/3=1==>10+3+1=14

I've seen this approach before by ywilfried. It's the best to use.
Re: PS factors and exponents.   [#permalink] 18 Jan 2006, 01:35
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