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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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29 Dec 2013, 03:09
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60% (01:35) correct 40% (01:28) wrong based on 303 sessions

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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
[Reveal] Spoiler: OA

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Last edited by Bunuel on 29 Dec 2013, 03:16, edited 1 time in total.
Renamed the topic and edited the question.

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29 Dec 2013, 03:10
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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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29 Dec 2013, 03:18
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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

Similar questions to practice:
if-12-16-12-8-12-16-12-4-a-what-is-the-unit-s-digit-86818.html
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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01 Aug 2015, 11:55
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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.

Ans 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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17 Nov 2016, 14:08
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

C.

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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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22 Jun 2017, 23:03
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

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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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26 Dec 2017, 11:02
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.

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Re: If a=(13!^1^6-13!^8)/(13!^8+13!^4) hat is the unit digit of   [#permalink] 26 Dec 2017, 11:02
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