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If p is a prime number, k is a positive integer, and \(p^{k}\) is a factor of \(20!\), what is the largest possible value of \(pk\) ?
(A) 36
(B) 48
(C) 53
(D) 64
(E) 72
(A) 36
(B) 48
(C) 53
(D) 64
(E) 72
Lenel
Joined: 21 Jan 2020
Last visit: 06 Jan 2024
Posts: 51
Location: India
Schools: ISB'22 IIMB EPGP'23 IIM-A '24
GMAT 1: 600 Q47 V25
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Given that p is prime and p^k is a factor of 20!,
20! has primes 2,3,5,7,11,13,17,19
and for p^k to be a factor of 20! it should be any one of 2^2, 2^3, 2^4, 3, 3^2, 5,7,11,13,17,19 where the powers represent k ie., k=1,2,3,4
Hence pk=9*4=36 (other options don't give the maximum value of pk)
Answer: A
20! has primes 2,3,5,7,11,13,17,19
and for p^k to be a factor of 20! it should be any one of 2^2, 2^3, 2^4, 3, 3^2, 5,7,11,13,17,19 where the powers represent k ie., k=1,2,3,4
Hence pk=9*4=36 (other options don't give the maximum value of pk)
Answer: A
Posts: 51
Ravixxx
I think the problem wholly is wrong or I am having some difficulty understanding . Bunuel can you have a look please .:
So 20! apparently consists of eight 3's (2 from 18,1 from each of 15,12,6,3 and 2 from 9 ) . So apparently 3^9 is a factor of 20! . Again 11^1 is also a factor . So this makes pk to be 99 .
I just realized if we take 2 and not 3 , then it will be far more. So am I thinking in the correct direction experts?
Updated on: 23 Apr 2020, 08:39
Originally posted by GMATinsight on 23 Apr 2020, 08:38.
Last edited by GMATinsight on 23 Apr 2020, 08:39, edited 1 time in total.
Last edited by GMATinsight on 23 Apr 2020, 08:39, edited 1 time in total.
Ravixxx
\(20! = 2^{18}*3^8*5^4*7^4*11^1*13^1*17^1*19^1\)
Now, If p = 2 then k = 18 and p*k = 36
Now, If p = 3 then k = 8 and p*k = 24
Now, If p = 5 then k = 4 and p*k = 20
Now we see a decreasing trend in values of p*k
i.e. \(p*k_{max.} = 36\)
Answer: Option A
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ElciaM
Hi ElciaM
Given that p = Prime, you can NOT choose the value of p = 9
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MrSengupta
Hi MrSengupta
Please note the fact that \(p^k\) is a factor of 20!
3^9 is NOT factor of 20! [Highlighted the mistake]
You can NOT choose p from someplace and k from some other factor. p^k must be factor of 20! so p and k both must belong to the same factor of 20!
I hope this help!!!
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GMATinsight
Yes thanks a lot . That helped .
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Ravixxx
for the given two conditions ;
a) P has to be prime and
b) find largest value of pk
the value of p has to be the least prime value i.e 2
so total factors of 20! at p =2 is ;
20/2+20/4+20/8+20/16 = 10+5+2+1 ; 18 = k
and p*k = 18*2 ; 36
OPTION A ;
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Akash: Avhad
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Primes that are factors of 20! include 2,3,5,7,11,13,17 and 19.
The highest powers of these primes that are also factors of 20! include 2^18, 3^8, 5^4, 7^2, 11^1, 13^1, 17^1 and 19^1
Any of these can be considered as p^k, but the one giving highest value of pk is 2^18.
In this case, pk = 2 x 18 = 36.
In all other cases, pk is less than 36.
Hence, answer is A.
Kudos if you liked the explanation.
Posted from my mobile device
The highest powers of these primes that are also factors of 20! include 2^18, 3^8, 5^4, 7^2, 11^1, 13^1, 17^1 and 19^1
Any of these can be considered as p^k, but the one giving highest value of pk is 2^18.
In this case, pk = 2 x 18 = 36.
In all other cases, pk is less than 36.
Hence, answer is A.
Kudos if you liked the explanation.
Posted from my mobile device
Gylmitul
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How did you decide that the minimum value will occur at the smallest prime factor?
I understand one way to do it is to observe the trend, but wanted to check if there is some other way also.
I understand one way to do it is to observe the trend, but wanted to check if there is some other way also.
Archit3110
Gylmitul
Hi Gylmitul
You have to maximize \(p^{k}\)
Thus in order to do that you have to choose the smallest prime factor but with the biggest k.
\(2^{18}\) is the best choice in order to maximize \(p^{k}\)
Gylmitul
Hi Gylmitul
One can NOT be sure about it in a short span of time (which we get while solving problem in test) so it's recommended that we see a pattern of how the values of the required function are changing rather than just staying with one set of values.
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Archit3110
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Gylmitul
well the question has asked for maximum value of p*k ; where p is prime integer and k is factor of 20 ! at p^k ;
determining that at which value of p will we get max k ; its basically the lower the value of divisor more is the share we get eg when 20/2 we get 10 parts and when 20/10 we get 2 parts.
similarly when we are finding the factors for 20! for a prime integer p ; the highest factors would be determined at lowest value p i.e 2 in this case..factors would be k
in this case
..
for this question we do need to determine the trend for atleast 2 different values of p .. you may do it for first two primes i.e 2 &3 and you would realise that p*k value is decreasing with every prime integer for 20! ..
once thats done then p*k can be found as done in explanation above...
hope this helps
well the question has asked for maximum value of p*k ; where p is prime integer and k is factor of 20 ! at p^k ;
determining that at which value of p will we get max k ; its basically the lower the value of divisor more is the share we get eg when 20/2 we get 10 parts and when 20/10 we get 2 parts.
similarly when we are finding the factors for 20! for a prime integer p ; the highest factors would be determined at lowest value p i.e 2 in this case..factors would be k
in this case
..
for this question we do need to determine the trend for atleast 2 different values of p .. you may do it for first two primes i.e 2 &3 and you would realise that p*k value is decreasing with every prime integer for 20! ..
once thats done then p*k can be found as done in explanation above...
hope this helps
Gylmitul
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Expert reply
Posts: 22,738
Ravixxx
For reference, here is 20! expanded: 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
The largest prime number that is a factor of 20! is 19. So p could be 19; however, in that case, k would be 1, and the value of pk would be 19(1) = 19. Now let’s see if we can find a value for pk that is greater than 19.
If p is any prime number from 11 to 17, inclusive, the value of k would still be 1, and the value of pk would then be less than 19.
If p = 7, then k = 2 (since 7 and 14 are factors of 20!). However, the value of pk is 7(2) = 14, which is less than 19.
If p = 5, then k = 4 (since 5, 10, 15 and 20 are factors of 20!). The value of pk is 5(4) = 20, which is greater than 19. However, 20 is not one of the answer choices, so we must continue.
If p = 3, then k = 8 (since 3, 6, 9, 12, 15, and 18 are factors of 20!. Notice that 9 and 18 have 2 factors of 3 while the other factors have 1 factor of 3 each). The value of pk is 3(8) = 24. Again, this is not one of the answer choices, so we must continue.
If p = 2, then k = 18 (since 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20 are factors of 20!. Notice that 4, 12, and 20 have 2 factors of 2, 8 has 3 factors of 2, and 16 has 4 factors of 2, while the other factors have 1 factor of 2 each). The value of pk is 2(18) = 36. Since we’ve exhausted all the possibilities, the largest value of pk is 36.
Answer: A
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Ravixxx
2 will have
2->10
4->5
8->2
16->1
Total= 18
Hence 2*18=36
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