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If x = a!, which of the following values of positive integer a is the

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Bunuel
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If x = a!, which of the following values of positive integer a is the [#permalink] New post   10 Jan 2017, 03:58
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If x = a!, which of the following values of positive integer a is the least possible value for which the last 8 digits of the integer x will all be zero?

A. 21
B. 29
C. 38
D. 40
E. 100
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C
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Re: If x = a!, which of the following values of positive integer a is the [#permalink] New post   10 Jan 2017, 04:48
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x = a! = (Some number)*10^8

10^8 = (5*2)^8 --> a! must contain 8 5's in it.

A. 21 --> 21/5 = 4
B. 29 --> 29/5 + 29/25 = 4 + 1 = 5
C. 38 --> 38/5 + 38/25 = 7 + 1 = 8
D. 40
E. 100

Answer: C
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Re: If x = a!, which of the following values of positive integer a is the [#permalink] New post   10 Jan 2017, 05:07
Bunuel wrote:
If x = a!, which of the following values of positive integer a is the least possible value for which the last 8 digits of the integer x will all be zero?

A. 21
B. 29
C. 38
D. 40
E. 100


By trailing zeroes formula, w.k.t., No of Trailing Zeroes of n!=n/5+n/5^2+...+n/5^k, where 5^k<n

The question asks us for the value of a when a! has 8 trailing zeroes.
By trial and error:

TZ. 38!= 38/5+38/25=7+1=8

Hence C.
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Re: If x = a!, which of the following values of positive integer a is the [#permalink] New post   10 Jan 2017, 05:50
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Bunuel wrote:
If x = a!, which of the following values of positive integer a is the least possible value for which the last 8 digits of the integer x will all be zero?

A. 21
B. 29
C. 38
D. 40
E. 100


To make 8 zeroes, we need 8 5s and 8 2s. The eight multiple of 5 is 40 but 25 already has an extra 5so we need to go only till the 7th multiple of 5 i.e. 35.
In 35!, we will have 8 5s and many more than 8 2s. So the last 8 digits will be all 0.
Of the given options, 38 is the smallest such number.

Answer (C)
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Re: If x = a!, which of the following values of positive integer a is the [#permalink] New post   18 Sep 2019, 11:10
Hi everyone,

I order to get the trailing zeros of a factorial number n!---> n!/5+ n!/5^2......
when analyzing C: 38/5+38/25=7+1=8
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Re: If x = a!, which of the following values of positive integer a is the [#permalink]
18 Sep 2019, 11:10
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