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If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di

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If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 27 Jun 2018, 21:29
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If \(\frac{(14!)^{25}-(14!)^{11}}{(14!)^{11}-(14!)^{4}}=X\).

What is the unit digit of\(\frac{X}{(14!)^7}\) ?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 27 Jun 2018, 22:08
PKN wrote:
If \(\frac{(14!)^{25}-(14!)^{11}}{(14!)^{11}-(14!)^{4}}=\)X,

What is the unit digit of\(\frac{X}{(14!)^7}\) ?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9


Similar questions to practice:
http://gmatclub.com/forum/if-12-16-12-8 ... 86818.html
http://gmatclub.com/forum/baker-s-dozen ... l#p1057520
https://gmatclub.com/forum/if-a-13-1-6- ... 65186.html

Hope this helps.
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 27 Jun 2018, 23:17
2

Solution



Given:
    • \(\frac{(14!)^{25}-(14!)^{11}}{(14!)^{11}-(14!)^{4}}=X\)

To find:
    • The unit digit of \(\frac{X}{(14!)^7}\)

Approach and Working:
Assuming n = 14!, we can rewrite the whole expression as
    • X = \(\frac{(n)^{25}-(n)^{11}}{(n)^{11}-(n)^{4}}\)
    = \(\frac{(n)^{11}[(n)^{14}-1]}{(n)^4[(n)^{7}-1}\)
    = \((n)^7\frac{(n^7+1)(n^7-1)}{(n^7-1)}\)
    = \(n^7 (n^7+1)\)

So, if we do \(\frac{X}{(14!)^7}\), we can write it as:
    • \(\frac{X}{(14!)^7}\) = \(\frac{n^7 (n^7+1)}{n^7}\) = \(n^7 + 1\)

As n = 14!, n will always end with a 0.
    • Therefore, the unit digit of \(n^7 + 1\) = 0 + 1 = 1

Hence, the correct answer is option A.

Answer: A
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 27 Jun 2018, 23:52
EgmatQuantExpert : how you have assumed that 14! will always end with a 0. It is mugged up or any logic behind it.
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 28 Jun 2018, 00:08
akshayvp wrote:
EgmatQuantExpert : how you have assumed that 14! will always end with a 0. It is mugged up or any logic behind it.


Let’s look at the factorial values of the following numbers:
    • 1! = 1
    • 2! = 1 * 2 = 2
    • 3! = 1 * 2 * 3 = 6
    • 4! = 1 * 2 * 3 * 4 = 24
    • 5! = 1 * 2 * 3 * 4 * 5 = 120

Now, if you go beyond 5!, in every factorial value 2 and 5 will always be present.

The product of 2 and 5 will always end with a 0, hence, 5! onwards every factorial value should always end with at least one 0.
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 28 Jun 2018, 00:22
akshayvp wrote:
EgmatQuantExpert : how you have assumed that 14! will always end with a 0. It is mugged up or any logic behind it.



14! = 14 * 13 * 12 * 11 * 10 * .....

You have to multiply by 10 in the equation above. As you know, any integer when multiplied by 10 , you add a zero at the end. Thats the logic used in the question above
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di  [#permalink]

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New post 29 Jun 2018, 04:22
Thanks for the explanation EgmatQuantExpert and pikolo2510
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Re: If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di &nbs [#permalink] 29 Jun 2018, 04:22
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