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

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Senior Manager
Joined: 18 Jun 2018
Posts: 263
What is the largest power of 6! that can divide 60! ?  [#permalink]

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13 Sep 2018, 10:02
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Difficulty:

55% (hard)

Question Stats:

56% (01:43) correct 44% (01:42) wrong based on 114 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
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 1043
Location: India
GPA: 3.64
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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13 Sep 2018, 10:30
3
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]
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Manager
Joined: 29 Sep 2016
Posts: 113
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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26 Sep 2018, 18:01
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.
Senior Manager
Joined: 18 Jun 2018
Posts: 263
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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27 Sep 2018, 21:10
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
Joined: 29 Sep 2016
Posts: 113
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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30 Sep 2018, 00:27
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.
Status: Preparing for GMAT
Joined: 25 Nov 2015
Posts: 1043
Location: India
GPA: 3.64
Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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30 Sep 2018, 02:28
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.
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Re: What is the largest power of 6! that can divide 60! ?  [#permalink]

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02 Oct 2018, 19:14
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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Re: What is the largest power of 6! that can divide 60! ?   [#permalink] 02 Oct 2018, 19:14
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