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# If x is a positive integer and 50!/30^x is a positive integer, what is

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Math Expert
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Posts: 64322
If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 00:14
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Competition Mode Question

If x is a positive integer and $$\frac{50!}{30^x}$$ is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 00:37
1
30=2*3*5
Our bottle neck is 5 so we need to know how many 5 are in 50!
we have for every 5 numbers 1 and on 25 and 50 we have 2 fives
thus 10+2=12
if we have 12 fives is 50! we can have 30^12 as the greatest value

Ans (b)
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 00:58
1
We need to Prime factorizing for 30 -> 2*3*5

So, the maximum number for 2 will be calculated by:
$$\frac{50}{2}$$ + $$\frac{50}{2^2}$$ + $$\frac{50}{2^3}$$ + $$\frac{50}{2^4}$$ + $$\frac{50}{2^5}$$ = 25+12+6+3+1= 47
So, the maximum number for 3 will be calculated by:
$$\frac{50}{3}$$ + $$\frac{50}{3^2}$$ + $$\frac{50}{3^3}$$ = 16+5+1= 22
So, the maximum number for 5 will be calculated by:
$$\frac{50}{5}$$ + $$\frac{50}{5^2}$$ = 10+2 = 12

As 30 is the combination of 2, 3 and 5 - We take the minimum value of them, i.e. - 12. 'B' is the winner
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 02:54
1
If x is a positive integer and $$\frac{50!}{30^{x} }$$ is a positive integer, what is the greatest possible value of x ?

$$30^{x}=2^{x}*3^{x}*5^{x}$$
--> In order to get the greatest possible value of x, finding the number of $$5s$$ in $$50!$$ is enough:

$$[\frac{50}{5}]+[\frac{50}{5^{2}}]+...= 10+2 =12$$ --> the number of $$5s$$ in $$50!$$
$$-->x = 12$$

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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 04:44
1
$$50!/(2^x . 3^x. 5^x)$$

The greatest value of positive integer x can be determined by following:
$$50/5 + 50/(5^2) + 50/(5^3) = 10 + 2 + 0 = 12$$

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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 06:31
1
30 = 2*3*5

Highest value is 5

So, 50!/5 = 10 & 10!/5 = 2

Hence, Answer must be 10 + 2 = 12, Answer must be (B) 12
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If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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Updated on: 26 Feb 2020, 00:05
1
If x is a positive integer and $$\frac{50!}{30^x}$$ is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

$$\frac{50!}{30^x}$$ = $$\frac{50!}{2^x*3^x*5^x}$$
Here x would take highest value from $$5^x$$ since $$2^x$$ would give value of x that would falsify the statement.
$$\frac{50}{5}$$ = 10 (highest power of 5 in 50!)
$$\frac{50}{25}$$ = 2 (highest power of 25 in 50!)
10 + 2 = 12
Hence highest power of 5 = 12.

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Originally posted by lnm87 on 25 Feb 2020, 07:24.
Last edited by lnm87 on 26 Feb 2020, 00:05, edited 1 time in total.
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 07:47
30^x = 5*2*3 ; to get highest possible value divide 50!/5 ;
50!/5 + 50!/25= 10+2; 12
IMO B

If x is a positive integer and 50!/30^x is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 08:23
1
I guess the answer is D.
I think that the exponent must comprise all the 3s that you can find from 1 to 30, which are progressively in the numbers 3,6(3x2),9(3^2),12(3x4),15(3x5),18(3^2x2),21(3x7),24(3x8),27(3^3) and finally 30(3x10).
In total they are 14, hence D.
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 16:17
1
50!/30^x

30=3*2*5.......so if we can get the no.of 5's in 50! Then that would be sufficient to find X....as 5 is the greatest prime factor of 5

50! Consists of 11 5's

So the max value of X is 11

OA:A
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 17:10
1
Quote:
If x is a positive integer and 50!/30^x is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

primefactors(30)=2,3,5
multiples of 5 in 50!: sum quotients 50/5+50/25=10+2=12

Ans (B)
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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25 Feb 2020, 23:55
1
30^x = (5*3*6)^x
Greatest possible value of x = 50/5 + 50/25
= 10 + 2 = 12
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is  [#permalink]

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28 Mar 2020, 03:29
Bunuel wrote:

Competition Mode Question

If x is a positive integer and $$\frac{50!}{30^x}$$ is a positive integer, what is the greatest possible value of x ?

A. 11
B. 12
C. 13
D. 14
E. 15

Are You Up For the Challenge: 700 Level Questions

Asked: If x is a positive integer and $$\frac{50!}{30^x}$$ is a positive integer, what is the greatest possible value of x ?

30 = 2*3*5
Power of 30 will depend on maximum power of 5 in 50!

Power of 5 in 50! = 10 + 2 = 12

IMO B
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Re: If x is a positive integer and 50!/30^x is a positive integer, what is   [#permalink] 28 Mar 2020, 03:29