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

 It is currently 21 Oct 2018, 01:09

### 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

# How many 4 digit numbers are there, if it is known that the

Author Message
TAGS:

### Hide Tags

Intern
Joined: 02 Nov 2009
Posts: 17
How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

28 Jan 2010, 15:45
3
25
00:00

Difficulty:

95% (hard)

Question Stats:

38% (02:34) correct 62% (02:35) wrong based on 434 sessions

### HideShow timer Statistics

How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?

A. 20
B. 150
C. 225
D. 300
E. 320
Math Expert
Joined: 02 Sep 2009
Posts: 50007

### Show Tags

28 Jan 2010, 16:19
14
6
How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?
A. 20
B. 150
C. 225
D. 300
E. 320

4 options for the first digit: 2, 4, 6, 8;
5 options for the second digit: 1, 3, 5, 7, 9;
4 options for the third digit: 2, 3, 5, 7;
4 options for the fourth digit: 0, 3, 6, 9.

Four digit # possible without the restriction (about the digit 2): 4*5*4*4=320

Numbers with two 2-s, 2X2X 1*5*1*4=20.

Thus there are 320-20=300 such numbers.

_________________
Manager
Joined: 08 Apr 2012
Posts: 122
Re: How many 4 digit numbers are there  [#permalink]

### Show Tags

31 May 2012, 01:23
5
1
Joy111 wrote:
How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?

A)20
B)150
C)225
D)300
E)320

Hi,

The first digit can be 2, 4, 6, 8.
The second digit can be 1, 3, 5, 7, 9
The third digit can be 2, 3, 5, 7
The fourth digit can be 0, 3, 6, 9

Case 1: Using 2 as the 1st digit only.

The first digit can be 2, 4, 6, 8. No. of selections = 4
The second digit can be 1, 3, 5, 7, 9. No. of selections = 5
The third digit can be 3, 5, 7. No. of selections = 3
The fourth digit can be 0, 3, 6, 9. No. of selections = 4

Total number of numbers = 4x5x3x4 = 240

Case 2: Using 2 as the 3rd digit only.

The first digit can be 4, 6, 8. No. of selections = 3
The second digit can be 1, 3, 5, 7, 9. No. of selections = 5
The third digit can be 2. No. of selections = 1
The fourth digit can be 0, 3, 6, 9. No. of selections = 4

Total number of numbers = 3x5x1x4 = 60

Hence, total number of numbers = 240 + 60 = 300

Regards,

Shouvik.
_________________

Shouvik
http://www.Edvento.com

##### General Discussion
Senior Manager
Joined: 25 Jun 2009
Posts: 279

### Show Tags

30 Jan 2010, 08:07
Bunuel wrote:
sudai wrote:
Hi All,

Need your help in solving the below problem!

How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?

The answer is 300, but why....

Thank you very much!!

4 options for the first digit: 2, 4, 6, 8;
5 options for the second digit: 1, 3, 5, 7, 9;
4 options for the third digit: 2, 3, 5, 7;
4 options for the fourth digit: 0, 3, 6, 9.

Four digit # possible without the restriction (about the digit 2): 4*5*4*4=320

Numbers with two 2-s, 2X2X 1*5*1*4=20.

Thus there are 320-20=300 such numbers.

Why do we have to consider 0 for the last digit? Shouldn't it be only 3,6, and 9 ?

Cheers
Intern
Joined: 02 Nov 2009
Posts: 17

### Show Tags

30 Jan 2010, 10:40
1
Because 0 is divisible by 3.
Manager
Joined: 12 May 2012
Posts: 71
Location: India
Concentration: General Management, Operations
GMAT 1: 650 Q51 V25
GMAT 2: 730 Q50 V38
GPA: 4
WE: General Management (Transportation)

### Show Tags

31 May 2012, 04:03
cipher wrote:

Why do we have to consider 0 for the last digit? Shouldn't it be only 3,6, and 9 ?

Cheers

I missed the '0'

hope to remember it in future
Intern
Joined: 19 Apr 2012
Posts: 25
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

25 Jul 2012, 10:56
"4 options for the first digit: 2, 4, 6, 8;"

Aren't there 5 options ? 0 is even as far as know and so meets also the condition ?
Director
Joined: 22 Mar 2011
Posts: 601
WE: Science (Education)
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

25 Jul 2012, 12:24
Trying to save some time...

For the second digit, 5 possibilities: 1, 3, 5, 7, 9
For the last (fourth) digit, 4 possibilities: 0, 3, 6, 9

Therefore, the total number of possibilities should be a multiple of 20 = 5 * 4.
A is out, being too small, there are more possibilities for the first and the third digit.

So, I have to chose between D and E.

First digit, 4 possibilities: 2, 4, 6, 8
Third digit also 4 possibilities: 2, 3, 5, 7

It would give 4 * 4 = 16, and 16 * 20 = 320, but because we have to allow only one digit of 2, the final number should be less than 320.
Only 300 is left.

Well, I guess being lazy to carry out all the computations it isn't always safe, but sometimes it feels so good to take even a little shortcut...
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Intern
Joined: 16 Apr 2012
Posts: 8
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

26 Jul 2012, 04:40
Alexmsi wrote:
"4 options for the first digit: 2, 4, 6, 8;"

Aren't there 5 options ? 0 is even as far as know and so meets also the condition ?

0 is even indeed.
But if you take 0 for your first digit, it means you have a 3digits number and we are looking for a 4 digits number.
Manager
Status: I will not stop until i realise my goal which is my dream too
Joined: 25 Feb 2010
Posts: 196
Schools: Johnson '15

### Show Tags

26 Jul 2012, 08:22
Bunuel wrote:
How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?
A. 20
B. 150
C. 225
D. 300
E. 320

4 options for the first digit: 2, 4, 6, 8;
5 options for the second digit: 1, 3, 5, 7, 9;
4 options for the third digit: 2, 3, 5, 7;
4 options for the fourth digit: 0, 3, 6, 9.

Four digit # possible without the restriction (about the digit 2): 4*5*4*4=320

Numbers with two 2-s, 2X2X 1*5*1*4=20.

Thus there are 320-20=300 such numbers.

_________________

Regards,
Harsha

Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

Satyameva Jayate - Truth alone triumphs

Intern
Joined: 15 Apr 2012
Posts: 8

### Show Tags

05 Aug 2012, 10:19
1
harshavmrg wrote:
Bunuel wrote:
How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?
A. 20
B. 150
C. 225
D. 300
E. 320

4 options for the first digit: 2, 4, 6, 8;
5 options for the second digit: 1, 3, 5, 7, 9;
4 options for the third digit: 2, 3, 5, 7;
4 options for the fourth digit: 0, 3, 6, 9.

Four digit # possible without the restriction (about the digit 2): 4*5*4*4=320

Numbers with two 2-s, 2X2X 1*5*1*4=20.

Thus there are 320-20=300 such numbers.

Restriction where the 4 digit number has two twos.
The two 2s can occur in a digit as shown by Bunuel- 2X2X
Therefore number of such numbers 1*5*1*4=20
Hope its clear now.
Director
Joined: 17 Dec 2012
Posts: 629
Location: India
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

17 Jul 2013, 00:58
1
sudai wrote:
How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?

A. 20
B. 150
C. 225
D. 300
E. 320

1. first digit is even - 2,4,6,8
2. second digit is odd - 1,3,5,7,9
3. third is prime - 2,3,5,7
4. fourth is divisible by 3- 0,3,6,9

In addition to the above 4 conditions we have the following:
digit 2 can be used only once

Assume two in (1) and (3) are both not used
the number of possibilities is 3*5*3*4=180
In addition to this either the two in (1) or the two in (3) is used
Number of possibilities if the two in (1) is used - 1*5*3*4=60
Number of possibilities if the two in (3) is used - 3*5*1*4=60
total number of possibilities= 180+60+60=300
_________________

Srinivasan Vaidyaraman
Sravna Holistic Solutions
http://www.sravnatestprep.com

Holistic and Systematic Approach

Manager
Joined: 18 Oct 2011
Posts: 87
Location: United States
Concentration: Entrepreneurship, Marketing
GMAT Date: 01-30-2013
GPA: 3.3
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

17 Jul 2013, 08:45
Funny how this is a 700 level question...not much to it really.

Scenario #1: Start with a 2 as the first number. This leaves 5,3 and 4 choices respectively for the rest. Total = 60 #'s
Scenario #2: Start with a 2 as the third number. This leaves 3,5 and 4 choices respectively for the rest. Total = 60 #'s
Scenario #3: Remove 2 as an option for the 1st and 3rd numbers. This leaves 3,5,3 and 4 choices respectively for the rest. Total = 180#'s

Add them up....60+60+180 = 300 Total numbers can be formed. (D)
Veritas Prep GMAT Instructor
Joined: 11 Dec 2012
Posts: 313
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

17 Jul 2013, 10:04
1
Since this has been solved a few times (all correctly), I'd like to spend a minute looking at the answer choices. This type of question is very hard to backsolve, but it's relatively easy to see where the trap answers lie.

How many 4 digit numbers are there, if it is known that the first digit is even, the second is odd, the third is prime, the fourth (units digit) is divisible by 3, and the digit 2 can be used only once?

A. 20
B. 150
C. 225
D. 300
E. 320

The correct answer is D, but if you neglect the restriction of 2 being used twice, the total number of options is E: 320. If you go the other way and over-emphasize this restriction, you end up with answer choice A: 20. C is the choice if you forget that zero is also a multiple of 3 (classic GMAT trap). You'd then have 3/4 as many choices and would get to C. B is a little harder to get to without making multiple mistakes, but it may be a fairly tempting number if you're guessing blindly.

It's obviously crucial to determine which answer choice is the correct one, but there is value in analyzing the other choices and seeing where the GMAT thinks your brain may go. Remember this exam is nothing if not foreseeable and preparable.

Hope this helps!
-Ron
_________________
Director
Joined: 23 Jan 2013
Posts: 575
Schools: Cambridge'16
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

30 Sep 2014, 07:42
Did

0,2,4,6,8
1,3,5,7,9
2,3,5,7 (removed 2 as condition says)
0,3,6,9

5*5*3*4=300

P.S. GMATprep considers 0 as even integer, so if use Bunuel's approach, the correct answer is 380
Intern
Joined: 30 Dec 2015
Posts: 4
How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

05 Dec 2016, 06:01
Hi,

I dont quite get why we dont use combination here. meaning there are 10 numbers for the first digit, but we can select only 4, hence 10C4. Hence, the total number of ways shouldnt be:

10C4 x 10C5 x 10C4 x 10C4

?

Can anyone explain to me?
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12687
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: How many 4 digit numbers are there, if it is known that the  [#permalink]

### Show Tags

06 Feb 2018, 12:03
Hi All,

This question has a lot of details to it, but the math involved isn't too bad (however, you will have to account for a series of 4-digit numbers that are NOT allowed, since they contain more than one 2 in their digits).

We're given a series of facts about the 4 digit number:
The first digit is EVEN: 2, 4, 6 or 8 (but not 0, since a 4-digit number can't start with 0)
The second digit is ODD: 1, 3, 5, 7, 9
The third digit is PRIME: 2, 3, 5, 7
The fourth digit is divisible by 3: 3, 6, 9, 0

If there were NO other restrictions, then the total number of 4-digit numbers would be:

(4)(5)(4)(4) = 320 options

However, there IS a restriction - the digit "2" can be used no more than once. Thus, any number that includes MORE than one 2 has to be removed... Thankfully, there aren't that many numbers that fit that description. If the first and third digits are both 2s, then the there are...

(1)(5)(1)(4) = 20 numbers with two 2s in the digits. Those 20 options have to be removed, which leaves us with...

320 - 20 = 300 4-digit numbers.

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****

Re: How many 4 digit numbers are there, if it is known that the &nbs [#permalink] 06 Feb 2018, 12:03
Display posts from previous: Sort by