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If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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09 Aug 2009, 12:10
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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 OPEN DISCUSSION OF THIS QUESTION IS HERE: ifpistheproductofintegersfrom1to30inclusive137721.html
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Last edited by Bunuel on 29 Sep 2014, 07:30, edited 1 time in total.
Renamed the topic, edited the question and added the OA.



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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09 Aug 2009, 14:02
I can see that any of these numbers is ok. so I just pick 18.
P = 30! = 30x29x28x......x4x3x2x1
if 3K is a factor of p, it means that we can divide p by 3K and the result is an integer. there is a "3" in the expression of p, and all numbers shown are also there.
the greatest one should then be OK.
is AO E??



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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09 Aug 2009, 23:04
k should be the largest prime <=30. So k can be 29 !! as 3 * 29 is a factor of 30!. However this option is not listed!!



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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09 Aug 2009, 23:22
Agree with Economist. But from the list, the best answer would be 18. Since all the prime factors of 18*3(2*3*3*3) would be in the product number. OA pls!
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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10 Aug 2009, 14:45
No....OA is C
please explain :s



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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tarek99 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
please show the fastest way to solve this. thanks Well ..I think question should read as 3^k ....i.e, 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? Then C ..14 will be correct..



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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11 Aug 2009, 04:32
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age wrote: tarek99 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
please show the fastest way to solve this. thanks Well ..I think question should read as 3^k ....i.e, 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? Then C ..14 will be correct.. I agree. If C is the OA, then the questions should read as 3^k giving 14 as the answer. Although if its not 3^k, then another interpretation of 3k could be 300 + k instead of 3*k, because 3*k makes the question silly. This means that when k=10, we get 310 and not 3*10 = 30 Thus proceeding with the "300 + k" assumption, we get: P = 30! Option A: 310 => 310 = 2 * 5 * 31 => 31 is a prime and is not present in 30! Thus 310 is not a factor of P => A is Eliminated Option B: 312 => 312 = 2^3 * 3 * 13 Thus 312 is a factor of P => B is the correct answer !Option C: 314 => 314 = 2 * 157 => 157 is a prime and is not present in 30! Thus 314 is not a factor of P => C is Eliminated Option D: 316 => 316 = 2^2 * 79 => 79 is a prime and is not present in 30! Thus 316 is not a factor of P => D is Eliminated Option E: 318 => 310 = 2 * 3 * 53 => 53 is a prime and is not present in 30! Thus 318 is not a factor of P => E is Eliminated ANS: B
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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30 Aug 2009, 00:18
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Factors of 31! include 14 times 3. Hence 3^14. Number: Factors of 3 3 : 1 6 : 1 9 : 2 12 : 1 15 : 1 18 : 2 21 : 1 24 : 1 27 : 3 30 : 1 Total : 14
Last edited by mendelay on 30 Aug 2009, 15:29, edited 1 time in total.



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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if it is 3*k, answer is E, just 18 in, 30! is divisible by 3*18 if it is 3^k, answer is C, 30!has 3, 6, 9,....30 as the factors, includes 14 of 3.
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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29 Sep 2014, 07:27
tarek99 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
please show the fastest way to solve this. thanks Can we change 3k to 3^K . A minor correction to make this question correct. Tx



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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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29 Sep 2014, 07:30
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tarek99 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 Finding the number of powers of a prime number k, in the n!.The formula is: \(\frac{n}{k}+\frac{n}{k^2}+\frac{n}{k^3}\) ... till \(n>k^x\) For 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: everythingaboutfactorialsonthegmat85592.html and mathnumbertheory88376.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. OPEN DISCUSSION OF THIS QUESTION IS HERE: ifpistheproductofintegersfrom1to30inclusive137721.html
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Re: If p is the product of the integers from 1 to 30, inclusive, what is t [#permalink]
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Please tag divisibility\factors\multiples Thanks tarek99 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 OPEN DISCUSSION OF THIS QUESTION IS HERE: ifpistheproductofintegersfrom1to30inclusive137721.html
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