GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 12 Dec 2019, 08:55 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # What is the sum of all 3 digit positive integers that can be formed us

Author Message
TAGS:

### Hide Tags

Manager  Joined: 31 May 2012
Posts: 108
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

8
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

One quickest way to answer this question !!

As we are using digits 1,5, 8 and digits are allowed to repeat. Each of the unit, tenth and hundredth digit can be used by each of three digits.
So, Total possible numbers with these digits=3 X 3 X 3 =27.

First, As we have 27 three digit number, Sum will be for sure more than 2700.. Eliminate options A,B,D Second, If you imagine numbers with the given digits 1,5,8. We have numbers like 888,885,855,858,851. Sum is for sure more than 4000. Eliminate option C. You are left with answer E.

----------------
consider giving a +kudo if this helps Intern  Joined: 11 Sep 2012
Posts: 7
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
1, 5 and 8 are allowed to be used 9 times as hundreds, tenths and units digit.

So you can line up:

9x100
9x 10
9x 1
9x500
9x 50
9x 5
9x800
9x 80
9x 8

When lining these up, you should quickly realize that it's bigger than 10.000 and pick your answer without going further.
MBA Section Director V
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 7331
City: Pune
What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

5
3
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

Answer to this question is easier to guess than to calculate. e.g. if we take 8 at hundreds place we would get at least 9 nos. So 800 * 9 = 7200 which surpasses every option but E.

Thru conventional method
(1+5+8)9 = 126
(1+5+8)9*10=1260
(1+5+8)9*100=12600

126 + 1260 + 12600 = 13986. E
_________________
2020 MBA Applicants: Introduce Yourself Here!

MBA Video Series - Video answers to specific components and questions about MBA applications.

2020 MBA Deadlines, Essay Questions and Analysis of all top MBA programs
Manager  Joined: 26 Sep 2013
Posts: 182
Concentration: Finance, Economics
GMAT 1: 670 Q39 V41 GMAT 2: 730 Q49 V41 Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

aismirnov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

Imagine, we have got all these possible numbers written down - there are in total 3^3 numbers (each digit can be either 1 or 5 or 8)

there are 3*3 options for having a number XY1
there are 3*3 options for having a number XY5
there are 3*3 options for having a number XY8

there are 3*3 options for having a number X1Z
there are 3*3 options for having a number X5Z
there are 3*3 options for having a number X8Z

there are 3*3 options for having a number 1YZ
there are 3*3 options for having a number 5YZ
there are 3*3 options for having a number 8YZ

we can sum units, tens and hundreds independently:
summing units gives (1+5+8)*3*3
summing tens gives (1+5+8)*10*3*3
summing hundreds gives (1+5+8)*100*3*3

How did you know how to do this? I mean, how did you learn? I have 4 weeks to go until my GMAT and I haven't gotten any better at these. I have no idea how to even begin approaching these problems, and none of these formulas make any sense to me.
Math Expert V
Joined: 02 Sep 2009
Posts: 59710
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
4
AccipiterQ wrote:
aismirnov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

Imagine, we have got all these possible numbers written down - there are in total 3^3 numbers (each digit can be either 1 or 5 or 8)

there are 3*3 options for having a number XY1
there are 3*3 options for having a number XY5
there are 3*3 options for having a number XY8

there are 3*3 options for having a number X1Z
there are 3*3 options for having a number X5Z
there are 3*3 options for having a number X8Z

there are 3*3 options for having a number 1YZ
there are 3*3 options for having a number 5YZ
there are 3*3 options for having a number 8YZ

we can sum units, tens and hundreds independently:
summing units gives (1+5+8)*3*3
summing tens gives (1+5+8)*10*3*3
summing hundreds gives (1+5+8)*100*3*3

How did you know how to do this? I mean, how did you learn? I have 4 weeks to go until my GMAT and I haven't gotten any better at these. I have no idea how to even begin approaching these problems, and none of these formulas make any sense to me.

Direct formulas are here: what-is-the-sum-of-all-3-digit-positive-integers-that-can-be-78143.html#p862674 Please ask if anything there is unclear.

Similar questions to practice:
find-the-sum-of-all-the-four-digit-numbers-formed-using-the-103523.html
find-the-sum-of-all-the-four-digit-numbers-which-are-formed-88357.html
find-the-sum-of-all-3-digit-nos-that-can-be-formed-by-88864.html
if-the-three-unique-positive-digits-a-b-and-c-are-arranged-143836.html
what-is-the-sum-of-all-3-digit-positive-integers-that-can-be-78143.html
what-is-the-sum-of-all-4-digit-numbers-that-can-be-formed-94836.html
the-sum-of-the-digits-of-64-279-what-is-the-141460.html
there-are-24-different-four-digit-integers-than-can-be-141891.html

Hope this helps.
_________________
Manager  Joined: 13 Apr 2015
Posts: 73
Concentration: General Management, Strategy
GMAT 1: 620 Q47 V28 GPA: 3.25
WE: Project Management (Energy and Utilities)
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
8
The Ans is E.

This can also be solved by using a formula

Sum of N numbers

With repetition = N ^ ( N - 1 ) * Sum of the Numbers * 111..... N no. of times

With out repetition = ( N - 1 ) ! * Sum of the Numbers * 111.... N no. of times.

Here repetition is allowed, therefore N is 3 ( No. of digits given ) and sum of No is 14.

3^2 * 14 * 111 = 13986.
Intern  Joined: 29 Oct 2015
Posts: 1
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

3
Used POE, No need to solve the whole question to get an exact value
Explanation:

since three digit numbers formed by 1, 5, 8 would be :
Lets start with numbers starting with 8 : 888, 885, 881, 855, 851, 815, 811
sum of these numbers is greater than Options A , B , C , D
Hence Ans: E.

Keep it simple ppl

Good luck!!
Cheers
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15704
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

6
Hi All,

There's a great 'pattern-matching' shortcut built into this question that can help you to avoid much of the 'math work' involved.

We're told to use the digits 1, 5 and 8 to form every possible 3-digit number (including those with duplicate digits) and then take the sum of those numbers.

Since the digits can be repeated, we're dealing with the numbers that fall into the range of 111 to 888, inclusive. There are (3)(3)(3) = 27 total numbers and 1/3 of those numbers will begin with an 8. From THAT deduction, we know that the sum of those 9 numbers will be greater than (9)(800) = 7200. There's only one answer that fits that description...

GMAT assassins aren't born, they're made,
Rich
_________________
Manager  B
Joined: 17 Jun 2015
Posts: 191
GMAT 1: 540 Q39 V26 GMAT 2: 680 Q46 V37 Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
With 1 in the hundreds place, there are 9 numbers that could be created, since repetition is allowed. So 1 occurs 9 times in hundreds place = 900
Similarly it occurs 9 times in tens place = 90
and 9 times in ones place = 9

999 * 1 = 999

Similarly for 5 and 8

999 ( 1 + 5 + 8) = 999 * 14 = (1000 - 1) * 14 = 14,000 - 14 = 13,986
_________________
Fais de ta vie un rêve et d'un rêve une réalité
Manager  Joined: 06 Jun 2014
Posts: 85
Location: United States
Concentration: Finance, General Management
GMAT 1: 450 Q27 V21 GPA: 3.47
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

2
(1+5+8) * 9 = 126
(1+5+8) * 9 * 10 = 1260
(1+5+8) * 9 * 10 * 10 =12600

Senior Manager  Joined: 23 Apr 2015
Posts: 279
Location: United States
WE: Engineering (Consulting)
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

2
R2I4D wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

I took a crude approach, the smallest number is 111 and largest is 888 and the number of integers are 3 * 3 * 3 = 27.
the smallest is 111 and largest is 888, so the mean is around 500, so 500*27 = 13,500.

+1 for kudos
Senior Manager  Joined: 23 Apr 2015
Posts: 279
Location: United States
WE: Engineering (Consulting)
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
sdrandom1 wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

There are in total 3 * 3 * 3 = 27 numbers in these combinations. The smallest is 111 and largest is 888, so the avg is approx, 500.
So the sum is approx 27 * 500 = 13,500 , close to E.
Intern  Joined: 29 May 2015
Posts: 10
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
2
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

Before solving, I knew that the number will be big but did not know how big.
As I did not know a fancy way of solving this, I just tried to find the number if the digits are NOT allowed to repeat.
Then it would be,

158
185
518
581
815
851

And just by rough estimation, it looked like it will be in 3000 range. As the question stem says digits ARE allowed to repeat, meaning,
there should be more than 3000. The only answer more than 3000 was E.
Intern  B
Joined: 27 Jan 2019
Posts: 8
What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

All these big formulas that people have used to solve this question are super impressive. However, it is much easier to answer this question using common sense and leveraging the answer choices.

If you simply write out the possibilities of three digit integers that would begin with an eight, such as 888, 818, 851, 815, 885, 858 (not sure if I'm missing some here), you'll see that the sum of just the 8s is greater than ~4800. Logically, therefore, once you take into account all the combinations, the number will be much bigger than ~4,800. All but one answer choice remains; therefore, E is the correct answer.
Intern  B
Joined: 01 Sep 2016
Posts: 7
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

1
Narenn wrote:
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

Answer to this question is easier to guess than to calculate. e.g. if we take 8 at hundreds place we would get at least 9 nos. So 800 * 9 = 7200 which surpasses every option but E.

Thru conventional method
(1+5+8)9 = 126
(1+5+8)9*10=1260
(1+5+8)9*100=12600

126 + 1260 + 12600 = 13896. E

The solution given has a typo error. It’s written as 13896, whereas it should be 13986
_________________
Best Regards,

Abhishek Roy
Math Expert V
Joined: 02 Sep 2009
Posts: 59710
Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

drroy wrote:
Narenn wrote:
asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

Answer to this question is easier to guess than to calculate. e.g. if we take 8 at hundreds place we would get at least 9 nos. So 800 * 9 = 7200 which surpasses every option but E.

Thru conventional method
(1+5+8)9 = 126
(1+5+8)9*10=1260
(1+5+8)9*100=12600

126 + 1260 + 12600 = 13896. E

The solution given has a typo error. It’s written as 13896, whereas it should be 13986

_____________________
Fixed the typo. Thank you.
_________________
Director  V
Joined: 24 Oct 2016
Posts: 586
GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 Re: What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

asimov wrote:
What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

Method 1: Sum Each Column of Digits

We need positive integers having 3 digits.

S = __ __ __

We can make 3*3*3 = 27 such positive integers since we can fill in each of the 3 spaces in 3 ways. Now imagine writing these 27 numbers one below the other to add.

158
185
...
...
x 27 combinations

When we add them, noticing the symmetry we know that there will be 9 1's in units digits, 9 5's and 9 8's. So units digits will add up to (1+5+8)*9.

Similarly, tens digits will add up (1+5+8)*9*10.
Similarly, hundreds digits will add up (1+5+8)*9*100

(1+5+8)*9 + (1+5+8)*9*10 + (1+5+8)*9*100 = (1+5+8) * 9 * (1+10+100) = 14 * 9 * 111 = 13,986

Method 2: Direct Formula

Sum of all n digit numbers formed by n non-zero digits with repetition being allowed is:

n^(n−1)∗(sum of the digits)∗(111... n times)

9 * 14 * 111 = 13,986

VP  P
Joined: 07 Dec 2014
Posts: 1217
What is the sum of all 3 digit positive integers that can be formed us  [#permalink]

### Show Tags

What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number?

A. 126
B. 1386
C. 3108
D. 308
E. 13986

111+555+888=1554 sum
1554/3=518 mean
3^3=27 numbers
27*518=13986 sum
E What is the sum of all 3 digit positive integers that can be formed us   [#permalink] 16 Nov 2019, 14:27

Go to page   Previous    1   2   [ 38 posts ]

Display posts from previous: Sort by

# What is the sum of all 3 digit positive integers that can be formed us  