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

Question

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

OptimusPrepJanielle · Accepted Answer

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

stonecold · Answer

Nice Question.
THe trick to be used here is 0! is actually 1

ronny123 · Answer

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

Abhishek009 · Answer

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.

abannore · Answer

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

QuantMadeEasy · Answer

0! =1

hence I and III 
D is correct

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

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