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# What is the last non-zero digit of expression

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Senior Manager
Joined: 18 Jun 2016
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GMAT 1: 720 Q50 V38
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What is the last non-zero digit of expression [#permalink]

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03 May 2017, 08:59
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95% (hard)

Question Stats:

28% (02:02) correct 73% (01:44) wrong based on 120 sessions

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What is the last non-zero digit of expression

$$\frac{(25!)}{(2 * 14!)}$$

Where a! = a * (a-1) * (a-2) * (a-3) * ... * 1

A. 2
B. 3
C. 4
D. 5
E. 6
[Reveal] Spoiler: OA

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Re: What is the last non-zero digit of expression [#permalink]

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03 May 2017, 10:29
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umg wrote:
What is the last non-zero digit of expression

$$\frac{(25!)}{(2 * 14!)}$$

Where a! = a * (a-1) * (a-2) * (a-3) * ... * 1

A. 2
B. 3
C. 4
D. 5
E. 6

Hi..

$$\frac{(25!)}{(2 * 14!)}=\frac{25*24*23*22*21*20*19*18*17*16*15}{2}$$..
25*15*16=$$5^3*2^4*3$$.. last non-0 digit is 6..
Let's check with last non-0 digit of remaining terms..
4*3*2*1*20*9*8*7/2=240*9*56=4*9*6=36*6=6*6=36..
So last digit =6..
E..

Otherwise if you know product of all digits, it will help
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Joined: 16 Feb 2017
Posts: 3
What is the last non-zero digit of expression [#permalink]

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03 Jun 2017, 09:34
chetan2u wrote:
Hi..

$$\frac{(25!)}{(2 * 14!)}=\frac{25*24*23*22*21*20*19*18*17*16*15}{2}$$..
25*15*16=$$5^3*2^4*3$$.. last non-0 digit is 6..
Let's check with last non-0 digit of remaining terms..
4*3*2*1*20*9*8*7/2=240*9*56=4*9*6=36*6=6*6=36..
So last digit =6..
E..

Otherwise if you know product of all digits, it will help

Hello,

Can you explain more please ?
This is too quick to be understandable by the beginners :/

I tried :

25!
--- = 25x24x23x21x20x19x18x17x8x15 The 8 is coming from 16/2 =8
(2x14!)

With that the last digit will be = 5x4x3x2x1x0X9x8x78x5 = 0
So now i have to do all the 25x24x23x21x20x19x18x17x8x15 to find something ?

Where am I wrong ?
Math Expert
Joined: 02 Aug 2009
Posts: 5715
Re: What is the last non-zero digit of expression [#permalink]

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04 Jun 2017, 07:51
Matth971 wrote:
chetan2u wrote:
Hi..

$$\frac{(25!)}{(2 * 14!)}=\frac{25*24*23*22*21*20*19*18*17*16*15}{2}$$..
25*15*16=$$5^3*2^4*3$$.. last non-0 digit is 6..
Let's check with last non-0 digit of remaining terms..
4*3*2*1*20*9*8*7/2=240*9*56=4*9*6=36*6=6*6=36..
So last digit =6..
E..

Otherwise if you know product of all digits, it will help

Hello,

Can you explain more please ?
This is too quick to be understandable by the beginners :/

I tried :

25!
--- = 25x24x23x21x20x19x18x17x8x15 The 8 is coming from 16/2 =8
(2x14!)

With that the last digit will be = 5x4x3x2x1x0X9x8x78x5 = 0
So now i have to do all the 25x24x23x21x20x19x18x17x8x15 to find something ?

Where am I wrong ?

Hi,

Remove EQUAL number of 5s and 2s as combined they will give you 0....
1) if there are more 5s than 2s, straight way the last digit will be 5.

Here 25 has 2*5s and 20 and 15 have one 5 each...
So 4*5s remove 4*2s also..
So 25*24*23*22*21*20*19*18*17*8*15 will become 24*23*22*21*4*19*18*17*8*3/(2*2*2*2)=3*23*22*21*2*19*18*17*8*3..
Now check for last digit..
4*3*2*1*4*9*8*7*8*3 =(4*3)*2*(9*8)*(4*3)*(8*7)...
Last digit in brackets are (2)*2*(2)*(2)*(6)=16*6..
Last digit is 6..

Hope it helps
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Joined: 03 Jun 2017
Posts: 1
What is the last non-zero digit of expression [#permalink]

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05 Jun 2017, 10:20
25! = 25*24*23*...*14!...
2*14!

When 25*24, or 21*20, or 20*19, or 16*15, these are the checkpoint and the last digit always comes to zero

After that, no matter what numbers multiply it, it should be zero too
0 * n = 0

When zero divided by 2, it becomes 5...

In this case, why the ans is not 5???

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Joined: 02 Aug 2009
Posts: 5715
Re: What is the last non-zero digit of expression [#permalink]

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05 Jun 2017, 10:29
windlouis wrote:
25! = 25*24*23*...*14!...
2*14!

When 25*24, or 21*20, or 20*19, or 16*15, these are the checkpoint and the last digit always comes to zero

After that, no matter what numbers multiply it, it should be zero too
0 * n = 0

When zero divided by 2, it becomes 5...

In this case, why the ans is not 5???

Posted from my mobile device

Posted from my mobile device

Hi,

It is not the case always..
It depends on the non zero digit next to zero..
1) if it's odd, then answer is 5.... 10/2=5 or 30/2=15
2) if it's even, it will depend on the digit... 20/2=10 or 40/2=20 and so on..

Hope it helps
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: What is the last non-zero digit of expression [#permalink]

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07 Mar 2018, 13:41
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$$\frac{(25!)}{(2∗14!)}$$

$$\frac{(25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15)}{2}$$

( 20 + 5) * ( 20 + 4) * ( 20 + 3) * ( 20 + 2) * ( 20 + 1) * 10 * ( 20 - 1) * ( 20 - 2) * ( 20 - 3) * ( 20 - 4) * (20 -5)

($$20^2$$ - $$5^2$$) * ($$20^2$$ - $$4^2$$) * ($$20^2$$ - $$3^2$$) * ($$20^2$$ - $$2^2$$) * ($$20^2$$ - $$1^2$$) * 10

( 400 - 25 ) * ( 400 - 16 ) * ( 400 - 9 ) * ( 400 - 4 ) * ( 400 - 1 ) * 10

so the unit digit will be formed from = 6 * 9 * 4 = 6
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Intern
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Posts: 16
Re: What is the last non-zero digit of expression [#permalink]

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07 Mar 2018, 15:07
Hi Chetan
Could you please elaborate your strategy in detail..I still did not get it. I have the exam day after tomorrow.. kind request
Thank you

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Intern
Joined: 20 Nov 2017
Posts: 16
Re: What is the last non-zero digit of expression [#permalink]

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07 Mar 2018, 15:08
Also, if possible give some example of similar question and how would you solve them

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Re: What is the last non-zero digit of expression   [#permalink] 07 Mar 2018, 15:08
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