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

 It is currently 16 Nov 2019, 17:26 ### 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.  # How many positive integers less than 10,000 are such that the product

Author Message
TAGS:

### Hide Tags

VP  Status: mission completed!
Joined: 02 Jul 2009
Posts: 1203
GPA: 3.77
How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

6
30 00:00

Difficulty:   95% (hard)

Question Stats: 22% (01:47) correct 78% (02:01) wrong based on 251 sessions

### HideShow timer Statistics

How many positive integers less than 10,000 are such that the product of their digits is 210?

A. 24
B. 30
C. 48
D. 54
E. 72

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 59086
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

5
5
How many positive integers less than 10,000 are such that the product of their digits is 210?
(A) 24
(B) 30
(C) 48
(D) 54
(E) 72

210=1*2*3*5*7=1*6*5*7. (Only 2*3 makes the single digit 6).

So, four digit numbers with combinations of the digits {1,6,5,7} and {2,3,5,7} and three digit numbers with combinations of digits {6,5,7} will have the product of their digits equal to 210.

{1,6,5,7} # of combinations 4!=24
{2,3,5,7} # of combinations 4!=24
{6,5,7} # of combinations 3!=6

24+24+6=54.

_________________
Retired Moderator Joined: 02 Sep 2010
Posts: 721
Location: London
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

11
3
Pkit wrote:
How many positive integers less than 10,000 are such that the product of their digits is 210?

24
30
48
54
72

210 = 2x5x3x7 = 5x6x7x1 = 5x6x7

Those are the only sets of digits we can use to for the numbers (any other combination of factors will have two digit factors).

Numbers using 2,5,3,7 = 4!
Numbers using 5,6,7,1 = 4!
Numbers using 5,6,7 (3-digit numbers) = 3!

_________________
##### General Discussion
Retired Moderator Joined: 20 Dec 2010
Posts: 1560
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

1
cvsmech wrote:
How many positive integers less than 10,000 are such that the product of their digits is 210?

a) 24
b) 30
c) 48
d) 54
e) 72.

Can anyone help ?

210=21*10=7*3*2*5

7325 can be arranged in 4! ways
3*2=6
765 can be arranged in 3! ways.
And
7651 can be arranged in 4! ways

Total=2*4!+3!=2*24+6=48+6=54

Ans: "D"
_________________
Intern  Joined: 30 Sep 2010
Posts: 46
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

less than 10,000 means it has to be less than 5 digits.

1) 4 digits
-------------
210 = 2*3*5*7 ... total 24 ways
210 = 1*6*5*7 ... total 24 ways

2) 3 digits
----------
6 * 5 * 7 .. total 6 ways.

hence total 54 ways...
Manager  Joined: 26 Jul 2010
Posts: 74
Location: India
Concentration: Operations, General Management
Schools: IIMA (M)
GMAT 1: 640 Q48 V29 GMAT 2: 670 Q49 V31 WE: Supply Chain Management (Military & Defense)
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

1. read question carefully--it says no $$< 10,000$$
that means $$10,000 < 4 digit > 999$$and 3 digit $$< 1000$$2. so now 210 has factors $$7,5,3,2,1$$

case 1 : four digit is possible with 7,5,3,2 because multiplication of digit$$=210$$
$$4!= 24$$
case 2 : we take 3X2=6 and then we can include 1 for four digit no, so no are 7,5,6,1
$$4!=24$$
Quote:
see other muliplication or cases cannot be included because multiplication goes to 2 digit no . ex $$7 X 3 =21$$. which is not possible

case 3 : 3 digit no, we can only take 7,6,5
so $$3!=6$$
adding all the case $$1,2,3= 54$$that is our answer D.
see gmat will not go complicate these kind of question further so all the best
_________________
lets start again
Manager  Joined: 23 Jan 2011
Posts: 98
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

How many positive integers less than 10,000 are such that the product of their digits is 210?

a) 24
b) 30
c) 48
d) 54
e) 72.

210 is the answer when 2, 3, 5 and 7 are multiplied. 210 can also be arrive using 5,6 and 7 and 1, 5, 6 and 7.

So sum of arrangements of 2357, 567 and 1567. This translates to 4! +3! + 4!, this equals to 24 + 6 + 24 = 54, D is the answer.
GMAT Tutor G
Joined: 24 Jun 2008
Posts: 1827
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

daviesj wrote:
how many positive integers less than 9999 are such that the product of their digits is 210.

A.24
B.30
C.48
D.56
E.72

Posted from my mobile device

The prime factorization of 210 is 2*3*5*7. So one way to make the right kind of number is to use those four digits, in any of the 4! = 24 orders you can put them in.

Notice though that we can also get 210 as a product by multiplying 5, 6 and 7. So we can make some 3-digit numbers with the right product: 3! = 6 of them to be exact.

But we can also get the right product using the digit 1 along with the digits 5, 6, and 7. Again we can arrange those digits in 4! = 24 orders.

So adding up the possible ways to make the right kinds of number, there are 24+24+6 = 54 ways. I think there might be a typo in your answer choices?
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Manager  Joined: 07 Apr 2014
Posts: 99
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

cvsmech wrote:
How many positive integers less than 10,000 are such that the product of their digits is 210?

A. 24
B. 30
C. 48
D. 54
E. 72

at first & got no answer for this question by missing the possibility for 3 digit number

prime factor of 210- 2 3 5 7 ... we could make 4! combination using this.. 6(2*3) , 5, 7 , 1 again 4! ... since number could be also three digits 6, 5 ,7 in 3!

4!+4!+3! =54
Intern  Joined: 06 May 2014
Posts: 11
how many positive integer less than 10,000 such that product of their  [#permalink]

### Show Tags

how many positive integer less than 10,000 such that product of their digits is 210?
ans- 54
how come??
Math Expert V
Joined: 02 Sep 2009
Posts: 59086
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

anik1989 wrote:
how many positive integer less than 10,000 such that product of their digits is 210?
ans- 54
how come??

Merging topics.

_________________
Math Expert V
Joined: 02 Aug 2009
Posts: 8176
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

anik1989 wrote:
how many positive integer less than 10,000 such that product of their digits is 210?
ans- 54
how come??

hi anik1989,
210= 2*3*5*7....
now there cannot be any two digits numbers satisfying the condition, as only one set of the numbers (2,3) will give you a single digit...
1) three digits number will consist of 6,5,7.. ways to arrange these three digits = 3!=6..
2) four digits number will consist of 2,3,5,7 or 1,6,5,7... each will have 4!=24.. TOTAL 24*2=48...
TOTAL 1+2=6+48=54... Hope it helped
_________________
Intern  Joined: 14 Feb 2015
Posts: 5
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

I love your sharing,i learn a lot from your post.There is a lot of very useful knowledge in your post.
Intern  Joined: 20 Jul 2016
Posts: 2
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

I have a question regarding this, I know it will be so basic, but I don't understand why 2x3x5x7x1 is not an option and 6x5x7x1 it is. I mean, why we count 1 in certain options and not in others?
Thanks.

Posted from my mobile device
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4064
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

Top Contributor
PTK wrote:
How many positive integers less than 10,000 are such that the product of their digits is 210?

A. 24
B. 30
C. 48
D. 54
E. 72

210 = (2)(3)(5)(7)

We need to consider 3 cases:

Case 1: 4-digit numbers using 2, 3, 5, 7
There are 4 digits, so this can be accomplished in 4! (24) ways

Aside: Notice that (2)(3) = 6

Case 2: 4-digit numbers using 1, 6, 5, 7
There are 4 digits, so this can be accomplished in 4! (24) ways

Case 3: 3-digit numbers using 6, 5, 7
There are 3 digits, so this can be accomplished in 3! (6) ways

Add up all 3 cases to get 24 + 24 + 6 = 54

RELATED VIDEO

_________________
Intern  Joined: 20 Jul 2016
Posts: 2
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

I'm sorry but I have to insist. I know that steps, but I don't know the reason why the '1' is not included in the first prime factorisation 2x3x5x7, and we include it in 6x5x7x1, making that different from 6x5x7 (the '1' gives us two different factorisation).
I don't know if I explain my request well.

Posted from my mobile device
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15461
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

Hi All,

To answer this question, you have to be very careful to be thorough (2 of the answers are "partial work" answers, meaning that could very easily get to one of those answers and think that you're correct, but you're not):

We're dealing with numbers less than 10,000 and we need the product of the digits to equal 210, so let's deal with THAT first:

210 = (2)(3)(5)(7)

With THOSE 4 digits (2,3,5,7), we could end up with 4x3x2x1 = 24 different numbers

But the question DID NOT state that the digits had to be prime, so that's NOT the only way to get to 210 using 4 digits:

We could use (1,5,6,7), and end up with 4x3x2x1 = 24 additional numbers

AND we have to consider numbers that are NOT 4-digits; we could use (5,6,7), which gives us 3x2x1 = 6 additional numbers

24 + 24 + 6 = 54

GMAT assassins aren't born, they're made,
Rich
_________________
Intern  Joined: 19 Jan 2019
Posts: 2
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

1B92 wrote:
I have a question regarding this, I know it will be so basic, but I don't understand why 2x3x5x7x1 is not an option and 6x5x7x1 it is. I mean, why we count 1 in certain options and not in others?
Thanks.

Posted from my mobile device

If you use 2 * 3 * 5 * 7 * 1 as 210's factorization , then you have to choose 4 different numbers from a pool of 5 numbers and create integers (between 0000 and 9999<10000) meaning 5P2 = 5*4*3*2 = 120 integers.

BUT

One such case is 7152. But 7*1*5*2 = 70 not 210. so you can't use that method.

You can only use 1 when 6 is used instead of 2*3. Because 6 * 5 * 7 * 1 = 210
Intern  Joined: 19 Jan 2019
Posts: 2
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

1B92 wrote:
I'm sorry but I have to insist. I know that steps, but I don't know the reason why the '1' is not included in the first prime factorisation 2x3x5x7, and we include it in 6x5x7x1, making that different from 6x5x7 (the '1' gives us two different factorisation).
I don't know if I explain my request well.

Posted from my mobile device

If you use 2 * 3 * 5 * 7 * 1 as 210's factorization , then you have to choose 4 different numbers from a pool of 5 numbers and create integers (between 0000 and 9999<10000) meaning 5P2 = 5*4*3*2 = 120 integers.

BUT

One such case is 7152. But 7*1*5*2 = 70 not 210. so you can't use that method.

You can only use 1 when 6 is used instead of 2*3. Because 6 * 5 * 7 * 1 = 210 Re: How many positive integers less than 10,000 are such that the product   [#permalink] 12 Feb 2019, 18:41
Display posts from previous: Sort by

# How many positive integers less than 10,000 are such that the product  