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If p is the product of integers from 1 to 30, inclusive [#permalink]
 23 Aug 2012, 11:18
Question Stats:
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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
Last edited by Bunuel on 12 Dec 2012, 05:36, edited 2 times in total.
Edited the question.
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Re: 3^k is a factor of p [#permalink]
23 Aug 2012, 20:27
If p is the product of 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 Values which we are looking for are 3,6,9,12,.. all multiples till 30 now everything will give you atleast 1 power of 3 but there are values which will give more than 1 power of 3 and those values will be multiples of 9 9 -> will give 2 powers 18-> will give 2 powers 27 -> will give 3 powers NUmber of single powers of 3 = (30-3)/3 +1 - 1(for 9) - 1(for 18) -1(for 27) = 7 so total powers = 2(for 9) + 2(for 18) + 3(for 27) + 7 = 14 Hope it helps!
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Re: 3^k is a factor of p [#permalink]
23 Aug 2012, 22:07
Hi nkdotgupta, Thanks for your reply but this is the explaation as provided in the OG. I wanted to know if there are other ways of approaching the problem since this method might be cumbersome we encounter larger numbers. nktdotgupta wrote: If p is the product of 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
Values which we are looking for are 3,6,9,12,.. all multiples till 30
now everything will give you atleast 1 power of 3 but there are values which will give more than 1 power of 3 and those values will be multiples of 9
9 -> will give 2 powers
18-> will give 2 powers
27 -> will give 3 powers
NUmber of single powers of 3 = (30-3)/3 +1 - 1(for 9) - 1(for 18) -1(for 27) = 7
so total powers = 2(for 9) + 2(for 18) + 3(for 27) + 7 = 14
Hope it helps!
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Re: 3^k is a factor of p [#permalink]
24 Aug 2012, 01:55
1
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ajju2688 wrote: If p is the product of 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 This is an OG12 question. Can someone tell me if there is a quick way to solve these kinds of questions since the OG explanation seems to be very time consuming? The formula is: \frac{n}{k}+\frac{n}{k^2}+\frac{n}{k^3} ... till n>k^xFor example: what is the power of 2 in 25! (the highest value of m for which 2^m is a factor of 25!) \frac{25}{2}+\frac{25}{4}+\frac{25}{8}+\frac{25}{16}=12+6+3+1=22. So the highest power of 2 in 25! is 22: 2^{22}*k=25!, where k is the product of other multiple of 25!. Check for more: everything-about-factorials-on-the-gmat-85592.html and math-number-theory-88376.htmlBack to the original question:If p is the product of 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 Given p=30!. Now, we should check the highest power of 3 in 30!: \frac{30}{3}+\frac{30}{3^2}+\frac{30}{3^3}=10+3+1=14. So the highest power of 3 in 30! is 1. Answer: C. Hope it's clear.
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Re: 3^k is a factor of p [#permalink]
24 Aug 2012, 02:26
Bunuel wrote: ajju2688 wrote: If p is the product of 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 This is an OG12 question. Can someone tell me if there is a quick way to solve these kinds of questions since the OG explanation seems to be very time consuming? The formula is: \frac{n}{k}+\frac{n}{k^2}+\frac{n}{k^3} ... till n>k^xFor example: what is the power of 2 in 25! (the highest value of m for which 2^m is a factor of 25!) \frac{25}{2}+\frac{25}{4}+\frac{25}{8}+\frac{25}{16}=12+6+3+1=22. So the highest power of 2 in 25! is 22: 2^{22}*k=25!, where k is the product of other multiple of 25!. Check for more: everything-about-factorials-on-the-gmat-85592.html and math-number-theory-88376.htmlBack to the original question:If p is the product of 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 Given p=30!. Now, we should check the highest power of 3 in 30!: \frac{30}{3}+\frac{30}{3^2}+\frac{30}{3^3}=10+3+1=14. So the highest power of 3 in 30! is 1. Answer: C. Hope it's clear. Thanks Bunuel for the crystal clear explanation! 
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Re: 3^k is a factor of p
[#permalink]
24 Aug 2012, 02:26
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