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Senior Manager  G
Joined: 18 Jun 2016
Posts: 257
Location: India
GMAT 1: 720 Q50 V38 GMAT 2: 750 Q49 V42 GPA: 4
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What is the last non-zero digit of expression  [#permalink]

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23 00:00

Difficulty:   95% (hard)

Question Stats: 27% (02:47) correct 73% (02:04) wrong based on 143 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

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Math Expert V
Joined: 02 Aug 2009
Posts: 7756
Re: What is the last non-zero digit of expression  [#permalink]

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2
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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Re: What is the last non-zero digit of expression  [#permalink]

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1
2
$$\frac{(25!)}{(2∗14!)}$$

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

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

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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 V
Joined: 02 Aug 2009
Posts: 7756
Re: What is the last non-zero digit of expression  [#permalink]

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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,

I'll take your method ahead..
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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What is the last non-zero digit of expression  [#permalink]

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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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Posted from my mobile device
Math Expert V
Joined: 02 Aug 2009
Posts: 7756
Re: What is the last non-zero digit of expression  [#permalink]

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

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

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Also, if possible give some example of similar question and how would you solve them

Sent from my SM-N910C using GMAT Club Forum mobile app
Senior Manager  G
Joined: 18 Jun 2016
Posts: 257
Location: India
GMAT 1: 720 Q50 V38 GMAT 2: 750 Q49 V42 GPA: 4
WE: General Management (Other)
Re: What is the last non-zero digit of expression  [#permalink]

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ra5867 wrote:
Also, if possible give some example of similar question and how would you solve them

Sent from my SM-N910C using GMAT Club Forum mobile app

Hey, I know that I'm very late for this but here it is for others who might need some more details. This is a self-made question. The strategy is simple.

Check the number of 5s and 2s. The following 3 cases are possible:

1. If there are more 5s than 2s, the Answer is 5 as an Odd number when multiplied by 5 give the unit digit 5.

2. If there are equal number of 5s and 2s, the answer is Odd but not 5.

3. If there are more 2s, which is the case here, the answer has to be Even. This eliminates 2 options.

Now, remove all the 5s and equal number of 2s and multiply for the Unit Digit.
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Re: What is the last non-zero digit of expression  [#permalink]

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_________________ Re: What is the last non-zero digit of expression   [#permalink] 01 May 2019, 09:59
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