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If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of

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If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of [#permalink]

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New post Updated on: 29 Dec 2013, 04:16
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If \(a=\frac{13!^1^6-13!^8}{13!^8+13!^4}\) what is the unit digit of \(\frac{a}{13!^4}\)?

A. 0
B. 1
C. 9
D. 4
E. 6

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Originally posted by gmat6nplus1 on 29 Dec 2013, 04:09.
Last edited by Bunuel on 29 Dec 2013, 04:16, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If [m]a=[fraction]13!^1^6-13!^8/13!^8+13!^4[/fraction][/m] w [#permalink]

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New post 29 Dec 2013, 04:10
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\(\frac{(13!^1^6-13!^8)}{(13!^8+13!^4)}=\frac{(13!^8-13!^4)(13!^8+13!^4)}{(13!^8+13!^4)}\). Cancel out the denominator of our fraction.

Now. \(\frac{a}{13!^4}=13!^4-1\). Unit digit of \(13!^4\) will always be 0. \(13!^4\) will always be greater than 1 and will have a unit digit of 0 let's assume it's 10. Now 10-1=9 which is our unit digit.
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Re: If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of [#permalink]

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New post 29 Dec 2013, 04:18
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gmat6nplus1 wrote:
If \(a=\frac{13!^1^6-13!^8}{13!^8+13!^4}\) what is the unit digit of \(\frac{a}{13!^4}\)?

A. 0
B. 1
C. 9
D. 4
E. 6


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baker-s-dozen-128782-40.html#p1057520

Hope this helps.
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Re: If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of [#permalink]

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New post 01 Aug 2015, 12:55
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After simplification, we have (13!)^4 - 1 = a/(13!)^4,

considering 13*12*11*10... 1, which has multiplier of 10, so the unit digit must end with 0 and if we subtract 1, the end digit will be 9.

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Re: If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of [#permalink]

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New post 17 Nov 2016, 15:08
Tricky little bugger.

Don't be fooled into thinking you need to multiply out the factorial. Nope.

1. Factor the equation & simplify
[(13!^8)(13!^8 -1)]/[(13!^4)(13!^4 +1)] --> [(13!^4)(13!^8 -1)]/[(13!^4 +1)]

2. We know a is going to be divided by 13!^4, so let's apply that to (1) as well
{[(13!^4)(13!^8 -1)]/[(13!^4 +1)]}/(13!^4)

We're left with 13!^4 -1 --> We know 13! will leave us with units digit of 0 (won't change if it's raised to a power). We need to subtract the 1 off a multiple of 10 and we will arrive at our answer.

10-1 = 9

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If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of [#permalink]

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New post 23 Jun 2017, 00:03
gmat6nplus1 wrote:
If \(a=\frac{13!^1^6-13!^8}{13!^8+13!^4}\) what is the unit digit of \(\frac{a}{13!^4}\)?

A. 0
B. 1
C. 9
D. 4
E. 6


\(a= \frac{13!^1^6-13!^8}{13!^8+13!^4}\)
\(=\frac{13!^8(13!^8-1)}{(13!^4(13!^4+1)}\)
\(=\frac{13!^4(13!^4+1)(13!^4-1)}{(13!^4+1)}\)
\(=13!^4(13!^4-1)\)
So, \(\frac{a}{13!^4} = (13!^4-1)\)

\(13!^4\) has unit's digit 0
so \((13!^4-1)\) has unit's digit = 10-1 =9


Answer C
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Re: If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of [#permalink]

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New post 26 Dec 2017, 12:02
In this problem... break the eqn into two solutions and you cancel the denominator.

13! is having 10 and hence, if 1 is subtracted from 13! we should be having 9 as a unit's digit.
Re: If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of   [#permalink] 26 Dec 2017, 12:02
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