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If n! and (n + 1)! have the same units digit, then n can be which of t

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If n! and (n + 1)! have the same units digit, then n can be which of t  [#permalink]

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New post 21 Apr 2016, 00:43
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A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

61% (00:54) correct 39% (00:58) wrong based on 144 sessions

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Re: If n! and (n + 1)! have the same units digit, then n can be which of t  [#permalink]

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New post 21 Apr 2016, 02:08
1
2
Bunuel wrote:
If n! and (n + 1)! have the same units digit, then n can be which of the following?

I. n = 10
II. n = 1
III. n = 0

A. I only
B. II only
C. III only
D. I and III
E. I, II, and III


Given: n! and (n + 1)! have the same units digits
Let us first write down the factorials of different numbers:

0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720

If we notice here, all the numbers after 4! will end with a 0.
Here is why:

If we multiply any number by 10, a 0 is added to that number
And a 10 is formed by a 5 and a 2, 10 = 5*2
Till 4!, we do not have a "5" in multiplication. But after that, we will have atleast one "5" in the multiplication.

Hence the units digits will be same after 5! and for 0! and 1!
Now checking the statements


I. n = 10
Yes, the units digits of 10! and 11! are same and = 0. TRUE

II. n = 1
units digit f 1! = 1 and units digit of 2! = 2. FALSE

III. n = 0
Units digit of 1! = 1 and units digit of 0! = 1. TRUE

Correct Option: D
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Re: If n! and (n + 1)! have the same units digit, then n can be which of t  [#permalink]

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New post 21 Apr 2016, 11:14
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Re: If n! and (n + 1)! have the same units digit, then n can be which of t  [#permalink]

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New post 21 Apr 2016, 17:14
2
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10!=3628800

All the numbers after 4! will end with a 0 as we have atleast one "5" in the multiplication.

Hence the units digits will be same after 5! and for 0! and 1!
Now checking the statements

I. n = 10
Yes, the units digits of 10! and 11! are same.
TRUE

II. n = 1
units digit f 1! = 1 and units digit of 2! = 2.
FALSE

III. n = 0
Units digit of 1! = 1 and units digit of 0! = 1.
TRUE

Correct Option: D
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Re: If n! and (n + 1)! have the same units digit, then n can be which of t  [#permalink]

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New post 22 Apr 2016, 13:15
Bunuel wrote:
If n! and (n + 1)! have the same units digit, then n can be which of the following?

I. n = 10
II. n = 1
III. n = 0

A. I only
B. II only
C. III only
D. I and III
E. I, II, and III


2 rules to make factorials easy

1. Any factorial greater than ( and including ) 5 will always have units digit as 0
2. 0! = 1! = 1


This question only tests the above two factorial rules...

Answer will definitely be (D)

PS : IMHO this problem can also be solved at a glance if the above properties are known to the test takers.
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Abhishek....

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Re: If n! and (n + 1)! have the same units digit, then n can be which of t  [#permalink]

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New post 04 Jan 2019, 21:33
Answer: D

(n+1)! = (n+1)n!
1. n=10
Since n! will include multiplication of all numbers from 1 to 10, the unit's digit will be a 0 (any number multiplied by 10 will have unit's digit 0). Similarly, (n+1)! will be multiplication of all numbers from 1 to 11. As this multiplication also includes 10, unit's digit will be 0.
Correct option

2. n=1
n!=1!=1
(n+1)!= 2!= 2
Incorrect option

3. n=0
n! = 0! = 1
(n+1)!= 1!= 1
Correct option
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Re: If n! and (n + 1)! have the same units digit, then n can be which of t   [#permalink] 04 Jan 2019, 21:33
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