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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, 22:29
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Question Stats:

61% (02:13) correct 39% (02:29) wrong based on 99 sessions

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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, 23:08
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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, 00:17
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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 28 Jun 2018, 00:52
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, 01:08
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, 01:22
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, 05:22
Thanks for the explanation EgmatQuantExpert and pikolo2510
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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 04 Feb 2019, 23:34
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


PLEASE

be careful while solving

So nothing you had to do a^2 - b^2 = (a-b) (a+b) & take the common out

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

take \((14!)^{11}\) common from numerator and \((14!)^{4}\) common from denominator

let 14! be x for sometime

x^11 - x^ 4 * [ x^14 - 1 / x^7 -1 ] = X

x/ x^7 = [ (x^7 - 1) (x^7+ 1) / x^7 -1 ]

x/x^7 = x^7 + 1

put back 14! in the above expression

we will certainly have a zero in the product of 14!

0+1

UD = 1

A
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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 05 Feb 2019, 16:11
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


My reasoning: set \(14! = a\)

Factor \(a^{11}\) out of the numerator and \(a^4\) out of the denominator. Since we're diving X by \(a^7\) we can cancel the \(a^{11}\) out of the numerator.

We're left with: \(\frac{(a^{14} -1)}{(a^7-1)}\)

We can cancel the denominator out and we're left with: \(a^7 +1\)

We know that the last digit in 14! will be 0 because there is a ten being multiplied. When we add one the final units digit will be 1. (answer choice A)
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If ((14!)^25 - (14!)^11)/((14!)^11 - (14!)^4) = X. What is the unit di   [#permalink] 05 Feb 2019, 16:11
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