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
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
Official Answer and Stats are available only to registered users. Register/ Login.




Math Expert
Joined: 02 Sep 2009
Posts: 50007

Re: Problem Solving
[#permalink]
Show Tags
28 Jan 2010, 16:19
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 2s, 2X2X 1*5*1*4=20. Thus there are 32020=300 such numbers. Answer: D.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Manager
Joined: 08 Apr 2012
Posts: 122

Re: How many 4 digit numbers are there
[#permalink]
Show Tags
31 May 2012, 01:23
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 Answer is D. Regards, Shouvik.
_________________
Shouvik http://www.Edvento.com admin@edvento.com




Senior Manager
Joined: 25 Jun 2009
Posts: 279

Re: Problem Solving
[#permalink]
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 2s, 2X2X 1*5*1*4=20. Thus there are 32020=300 such numbers. Why do we have to consider 0 for the last digit? Shouldn't it be only 3,6, and 9 ? Please explain. Cheers



Intern
Joined: 02 Nov 2009
Posts: 17

Re: Problem Solving
[#permalink]
Show Tags
30 Jan 2010, 10:40
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)

Re: Problem Solving
[#permalink]
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 ?
Please explain.
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

Re: Problem Solving
[#permalink]
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 2s, 2X2X 1*5*1*4=20.
Thus there are 32020=300 such numbers.
Answer: D. Hello Bunuel, can you please help me understand how u calculated for the restriction?
_________________
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

Re: Problem Solving
[#permalink]
Show Tags
05 Aug 2012, 10:19
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 2s, 2X2X 1*5*1*4=20.
Thus there are 32020=300 such numbers.
Answer: D. Hello Bunuel, can you please help me understand how u calculated for the restriction? 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
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: 01302013
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
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 overemphasize 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
_________________
Ron Awad Veritas Prep  GMAT Instructor Save $100 on Veritas Prep GMAT Courses and Admissions Consulting Services Veritas Prep Reviews



Director
Joined: 23 Jan 2013
Posts: 575

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/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12687
Location: United States (CA)

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 4digit 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 4digit 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 4digit 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 4digit numbers. Final Answer: 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
CoFounder & 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






