It is currently 21 Oct 2017, 10:54

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If n is a positive integer, what is the units digit of the sum of the

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 41894

Kudos [?]: 129127 [0], given: 12194

If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

08 Jul 2015, 03:40
Expert's post
10
This post was
BOOKMARKED
00:00

Difficulty:

85% (hard)

Question Stats:

42% (01:35) correct 58% (01:19) wrong based on 178 sessions

### HideShow timer Statistics

If n is a positive integer, what is the units digit of the sum of the following series:1! + 2! + ... + n!? (The series includes every integer between 1 and n, inclusive)

(1) n is divisible by 4.
(2) n^2 + 1 is an odd integer.

Kudos for a correct solution.
[Reveal] Spoiler: OA

_________________

Kudos [?]: 129127 [0], given: 12194

Math Forum Moderator
Joined: 20 Mar 2014
Posts: 2675

Kudos [?]: 1725 [3], given: 792

Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

08 Jul 2015, 04:37
3
KUDOS
Expert's post
1
This post was
BOOKMARKED
Bunuel wrote:
If n is a positive integer, what is the units digit of the sum of the following series:1! + 2! + ... + n!? (The series includes every integer between 1 and n, inclusive)

(1) n is divisible by 4.
(2) n^2 + 1 is an odd integer.

Kudos for a correct solution.

We know 1!=1, 2!=2, 3!=6, 4!=24, 5!=120 etc with emphasis on 5! and greater have 0 as the unit's digit.

1!+2!+3!+4! = 33 with unit's digit =3.

1. Per statement 1, n=4p.

If n=4, unit digit =3

If n =8 or 12 or 16 or 20 , unit digit =3 again as all factorials from >=5 have 0 as the unit's digit (as mentioned above). Thus this statement is sufficient.

2. Per statement 2, n^2+1=odd --> n^2=even --> n=even integer

n=2, unit digit =3
n=4 . unit digit =3 and
finally n=6 and above , unit digit is 0

Thus for this case as well unit digit = 3 and is thus sufficient.

Ans is D, both statements are sufficient to answer this question.
_________________

Thursday with Ron updated list as of July 1st, 2015: http://gmatclub.com/forum/consolidated-thursday-with-ron-list-for-all-the-sections-201006.html#p1544515
Inequalities tips: http://gmatclub.com/forum/inequalities-tips-and-hints-175001.html
Debrief, 650 to 750: http://gmatclub.com/forum/650-to-750-a-10-month-journey-to-the-score-203190.html

Kudos [?]: 1725 [3], given: 792

Current Student
Joined: 21 Aug 2014
Posts: 138

Kudos [?]: 225 [3], given: 49

GMAT 1: 610 Q49 V25
GMAT 2: 730 Q50 V40
Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

08 Jul 2015, 05:31
3
KUDOS
1
This post was
BOOKMARKED
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
From 5! onwards, all the factorials will end in 0, as they include a factor of 10. (5! = 5*4*2*1)

1!+2! = 1+ 2 = 3
1! + 2!+3! = 1+2+6=9
1! + 2!+3! + 4! = 1+2+6+24=33 = ...3
1! + 2!+3! + 4! + 5! = 1+2+6+24+120 =153 = ...3
Now since all the higher factors contain 0 as its units digit, the unit digit of their sum will not be affected by further addition.
So, the threshold figure is 4.

Statement (1) n is divisible by 4.
From the explanation above, the unit digit will always be 3. Sufficient.

Statement (2) $$n^2 + 1$$ is an odd integer.
$$n^2 + 1$$= Odd
$$n^2$$ = Even
$$n$$= Even
So, $$n$$= 2, 4, 6..
$$n$$=2; 1!+2! = 3
And from 4 onwards, the unit digit will always be 3.
In both the cases ($$n$$=2, even integer >2), the unit digit is 3.
Sufficient.

_________________

Please consider giving Kudos if you like my explanation

Kudos [?]: 225 [3], given: 49

Joined: 24 Oct 2012
Posts: 194

Kudos [?]: 115 [1], given: 45

Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

08 Jul 2015, 18:50
1
KUDOS
Bunuel wrote:
If n is a positive integer, what is the units digit of the sum of the following series:1! + 2! + ... + n!? (The series includes every integer between 1 and n, inclusive)

(1) n is divisible by 4.
(2) n^2 + 1 is an odd integer.

Kudos for a correct solution.

To find the Unit digit of 1! + 2! + ... + n! we need to find the unit digit of each component.

Unit digit of n! where n > 4 will be equal to Zero (since it will have component of 5 and 2). So any addition to series above 4 (n>4) won't impact unit digit.

Unit digit of series will be some thing like below
1! = 1
1! + 2! = 3
1! + 2! + 3! = 9
1! + 2! + 3! + 4! = 3
1! + 2! + 3! + 4! + 5! = 3 (Here on unit digit will be 3 only)

So we need to know, if n = 1,2,3 or n >= 4

Statement 1: n is divisible by 4. So lowest possible value of n = 4. So unit digit will be 3 (See above explanation)

Hence Sufficient

Statement 2 : n^2 + 1 is an odd integer

n^2 - is even integer
n is even integer, So possible value of n = 2,4,6,8
So in each case unit digit is 3

Hence Option D

Kudos [?]: 115 [1], given: 45

SVP
Joined: 08 Jul 2010
Posts: 1836

Kudos [?]: 2279 [0], given: 51

Location: India
GMAT: INSIGHT
WE: Education (Education)
Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

08 Jul 2015, 23:58
Bunuel wrote:
If n is a positive integer, what is the units digit of the sum of the following series:1! + 2! + ... + n!? (The series includes every integer between 1 and n, inclusive)

(1) n is divisible by 4.
(2) n^2 + 1 is an odd integer.

Kudos for a correct solution.

1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720 and so on

While we look at the series we understand that Factorial of every value greater than 4 has unit digit 0

i.e. 1! + 2! + ... + n! will not given any different value of Unit Digit for any value of n>4 for

Statement 1: n is divisible by 4

i.e. n can be any one of 4, 8, 12, 16,... etc.

The Unit digit of given expression will always be 1+2+6+4 = 3 (Unit digit) because every Factorial value greater than 4! will have unit digit 0 and won't influence the result
SUFFICIENT

Statement 2: n^2 + 1 is an odd integer
i.e. n^2 is Even
i.e. n may be 2 or 4 or 6 etc.
@n=2, the unit digit of 1!+2! = 3
@n=4, the unit digit of 1!+2!+3!+4! = 3
and every Factorial value greater than 4! will have unit digit 0 and won't influence the result
SUFFICIENT

_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Kudos [?]: 2279 [0], given: 51

Math Expert
Joined: 02 Sep 2009
Posts: 41894

Kudos [?]: 129127 [0], given: 12194

Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

13 Jul 2015, 03:02
Expert's post
1
This post was
BOOKMARKED
Bunuel wrote:
If n is a positive integer, what is the units digit of the sum of the following series:1! + 2! + ... + n!? (The series includes every integer between 1 and n, inclusive)

(1) n is divisible by 4.
(2) n^2 + 1 is an odd integer.

Kudos for a correct solution.

800score Official Solution:

In this question, we need to look for a pattern in the units digit of the sum of factorials for positive integers.
Then since 3! = 6 add this to the previous sum and get 9.
4! = 24. Add this to the previous sum: 33.
5! = 120. Adding this: 153.
6! = 720…

Let’s stop here, because the pattern should now be clear. Every additional number that will be added to the series will have a units digit of zero (because it has 2 and 5 as factors and therefore 10).

So for all n > 3, the units digit of the sum will be 3. Now, let's look at the statements.

Statement (1) tells us that n is a multiple of 4. Since n must be greater than 3, we know the units digit of the sum of the series will always be 3, so Statement (1) is sufficient.

From Statement (2), we can determine that n cannot be either 1 or 3. It is possible that n = 2, in which case: 1! + 2! = 3. Any other number that satisfies the statement is greater than 4, in which case the sum 1! + ... + n! will have a units digit of 3 as well. So Statement (2) is also sufficient because the units digit is always 3.

Since both statements are sufficient individually, the correct answer is choice (D).
_________________

Kudos [?]: 129127 [0], given: 12194

Manager
Joined: 08 Jun 2015
Posts: 123

Kudos [?]: 44 [0], given: 40

If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

20 Jul 2015, 09:40
The sum always ends in 3.

Before even looking at (1) or (2)....
1*2=2 *3=6 *4=24 *5... focus only on the units digit from now on... 4*5 = 20 .... 20*6 = 120 all factorials beyond 4 will always end in 0, because 0 times anything is always 0. Summing up the factorials before the 0 units digit factorials we get:

1+2+6+24 = 33. Adding 33 to any factorial where n = 5 or greater will result in a units digit of 3, because 3+0+0+0+0+0+0+0+... will always be 3.

Therefore, we know that all units digits for the sum of n>=4 will always be 3.

(1) n is a multiple of 4, so n = 4k. N = 4, 8, 12, 16, etc. Since n >= 4, n must have a units of 3.
Sufficient.

(2) (2) n^2 + 1 is an odd integer.
This tells us that n is even.
All positive, even integers equal to 4 or greater must have a units digit of 3. How about 2? The sum of factorials where n = 2 is also 3, since 2! + 1! = 3. Therefore, all even values of n will have a units digit of 3 when we sum up the given factorial series.
Sufficient.

Kudos [?]: 44 [0], given: 40

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16589

Kudos [?]: 273 [0], given: 0

Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

25 Aug 2016, 09:53
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 273 [0], given: 0

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16589

Kudos [?]: 273 [0], given: 0

Re: If n is a positive integer, what is the units digit of the sum of the [#permalink]

### Show Tags

18 Sep 2017, 12:28
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 273 [0], given: 0

Re: If n is a positive integer, what is the units digit of the sum of the   [#permalink] 18 Sep 2017, 12:28
Display posts from previous: Sort by