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Manager  D
Joined: 18 Jun 2018
Posts: 247
What is the largest power of 6! that can divide 60! ?  [#permalink]

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12 00:00

Difficulty:   55% (hard)

Question Stats: 56% (01:46) correct 44% (01:48) wrong based on 124 sessions

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What is the largest power of 6! that can divide 60!?

A) 56
B) 42
C) 28
D) 14
E) 7
Retired Moderator V
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 1039
Location: India
GPA: 3.64
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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4
2
Bismarck wrote:
What is the largest power of 6! that can divide 60!?

A) 56
B) 42
C) 28
D) 14
E) 7

6!=6.5.4.3.2.1
6=3x2
The greatest prime in 6! is 5, so finding the maximum power of 5 in 60!
$$\frac{60}{5}+\frac{60}{{5^2}}$$ = 12+2 = 14 [Ignoring the remainders]
##### General Discussion
Manager  B
Joined: 29 Sep 2016
Posts: 110
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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souvonik2k wrote:
Bismarck wrote:
What is the largest power of 6! that can divide 60!?

A) 56
B) 42
C) 28
D) 14
E) 7

6!=6.5.4.3.2.1
6=3x2
The greatest prime in 6! is 5, so finding the maximum power of 5 in 60!
$$\frac{60}{5}+\frac{60}{{5^2}}$$ = 12+2 = 14 [Ignoring the remainders]

Hi,
6fac has 2 to the power 4.

We need to find out how many 2 to the power 4 are present in 60fac.

60/2 = 30
60/4 = 15
60/8 = 7
60/16 = 3
60/32 = NIL

Total 55 2's.
55/4(2 is to the power of 4) = 13 (ignoring remainders)

Hence the answer must be 13 not 14.
Manager  D
Joined: 18 Jun 2018
Posts: 247
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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AKY13

Quote:
We need to find out how many 2 to the power 4 are present in 60fac.

60/2 = 30
60/4 = 15
60/8 = 7
60/16 = 3
60/32 = NIL 1

60/32 should be 1
Manager  B
Joined: 29 Sep 2016
Posts: 110
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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Bismarck wrote:
AKY13

Quote:
We need to find out how many 2 to the power 4 are present in 60fac.

60/2 = 30
60/4 = 15
60/8 = 7
60/16 = 3
60/32 = NIL 1

60/32 should be 1

Thanks for pointing out. However the answer is same, I think finding out the no. of 5s in 60 fac is not the correct way of doing. 2^4 is larger than 5 hence would be less in nos.
Retired Moderator V
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 1039
Location: India
GPA: 3.64
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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AKY13 wrote:
Bismarck wrote:
AKY13

Quote:
We need to find out how many 2 to the power 4 are present in 60fac.

60/2 = 30
60/4 = 15
60/8 = 7
60/16 = 3
60/32 = NIL 1

60/32 should be 1

Thanks for pointing out. However the answer is same, I think finding out the no. of 5s in 60 fac is not the correct way of doing. 2^4 is larger than 5 hence would be less in nos.

Hi
Similar question for practice:
https://gmatclub.com/forum/what-is-the- ... fl=similar
For theory you can refer to the following blog:
https://www.veritasprep.com/blog/2011/0 ... actorials/
Hope it helps.
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 11099
Location: United States (CA)
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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Bismarck wrote:
What is the largest power of 6! that can divide 60!?

A) 56
B) 42
C) 28
D) 14
E) 7

We note that 6! = 6 x 5 x 4 x 3 x 2 x 1 = 5^1 x 3^2 x 2^4 = 16 x 9 x 5. We need to determine the greatest power of these factors present in 60!.

To determine the number of 16s within 60!, we need to determine the number of 2s. We can use the following shortcut in which we divide 60 by 2, then divide the quotient of 60/2 by 2 and continue this process until we no longer get a nonzero quotient.

60/2 = 30

30/2 = 15

15/2 = 7 (we can ignore the remainder)

7/2 = 3 (we can ignore the remainder)

3/2 = 1 (we can ignore the remainder)

Since 1/2 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 2 within 60!.

Thus, there are 30 + 15 + 7 + 3 + 1 = 56 factors of 2 within 60!. Since 16 = 2^4, the largest power of 16 that divides 60! is 14.

Doing the same for the prime factor of 3 to determine the number of 9s in 60!, we find that the largest power of 9 that divides 60! is also 14.

Finally, since there are 12 multiples of 5 in 60! and both 25 and 50 contains two 5’s apiece, we see that if we were to break 60! into primes, we’d have 14 factors of 5. Thus, the largest power of 6! that divides 60! is 14.

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IESE School Moderator S
Joined: 11 Feb 2019
Posts: 308
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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I solved this using number of 0's in 60!
6! can be written as 2^4*3^2*5 : It will has only 0 which depends on power of 5
So i counted that 60! will have 5^14

so 14
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NJ Re: What is the largest power of 6! that can divide 60! ?   [#permalink] 28 May 2020, 13:35

# What is the largest power of 6! that can divide 60! ?  