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What is the greatest value of n such that 18^n is a factor of 18! ?
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20 Mar 2023, 01:30
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What is the greatest value of n such that 18^n is a factor of 18! ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Re: What is the greatest value of n such that 18^n is a factor of 18! ?
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10 May 2024, 11:45
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answer is D, 4
18 can be written as 2*3^2
18^n= 2^n*3^2n
following the factorial manipulation method here https://gmatclub.com/forum/variations-i ... l#p2189076
max power of 2 divisible by 18! is 16 = 9+4+2+1
18/2= 9
18/4=4
18/8=2
18/16=1
max power of 3 divisible by 18! is 8= 6+2
18/3=6
18/9= 2
(any higher power will give value<1 so we stop here)
=> max power of 3 is the constraint here, max power of 3 is 8 , 3^8 written as 3^2(4)
i.e n=4
hope this helps!
18 can be written as 2*3^2
18^n= 2^n*3^2n
following the factorial manipulation method here https://gmatclub.com/forum/variations-i ... l#p2189076
max power of 2 divisible by 18! is 16 = 9+4+2+1
18/2= 9
18/4=4
18/8=2
18/16=1
max power of 3 divisible by 18! is 8= 6+2
18/3=6
18/9= 2
(any higher power will give value<1 so we stop here)
=> max power of 3 is the constraint here, max power of 3 is 8 , 3^8 written as 3^2(4)
i.e n=4
hope this helps!
General Discussion
Re: What is the greatest value of n such that 18^n is a factor of 18! ?
[#permalink]
20 Mar 2023, 01:44
2
Kudos
18^n
2 and 3^2 prime factors
so in 18!
max power of 3 is 8
so n will be 4 and greatest power to make 18^n factor of 18!
option D
2 and 3^2 prime factors
so in 18!
max power of 3 is 8
so n will be 4 and greatest power to make 18^n factor of 18!
option D