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

 It is currently 15 Feb 2019, 17:18

### 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 February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### \$450 Tuition Credit & Official CAT Packs FREE

February 15, 2019

February 15, 2019

10:00 PM EST

11:00 PM PST

EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth \$100 with the 3 Month Pack (\$299)
• ### Free GMAT practice

February 15, 2019

February 15, 2019

10:00 PM EST

11:00 PM PST

Instead of wasting 3 months solving 5,000+ random GMAT questions, focus on just the 1,500 you need.

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 52905
How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 02:51
2
6
00:00

Difficulty:

95% (hard)

Question Stats:

30% (02:31) correct 70% (02:33) wrong based on 128 sessions

### HideShow timer Statistics

How many positive integers less than 10,000 are such that the product of their digits is 30?

A. 12
B. 24
C. 36
D. 38
E. 50

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 52905
How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

25 May 2015, 07:21
Bunuel wrote:
How many positive integers less than 10,000 are such that the product of their digits is 30?

A. 12
B. 24
C. 36
D. 38
E. 50

OFFICIAL SOLUTION:

30 = 2*3*5 = 6*5 (only 2*3 gives single digit number 6).

So, we should count the number of positive integers less than 10,000 with the digits {2, 3, 5} and {5, 6} and any number of 1's with each set.

2-digit numbers:
{5, 6} - the number of combinations = 2: 56 or 65.

3-digit numbers:
{1, 5, 6} - the number of combinations = 3! = 6: 156, 165, 516, 561, 615, or 651.
{2, 3, 5} - the number of combinations = 3! = 6.

4-digit numbers:
{1, 1, 5, 6} - the number of combinations = 4!/2! = 12.
{1, 2, 3, 5} - the number of combinations = 4! = 24.

Total = 2 + 6 + 6 + 12 + 24 = 50.

_________________
##### General Discussion
Manager
Joined: 12 Nov 2014
Posts: 63
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

Updated on: 20 May 2015, 08:14
1
2
30 = 1 * 30, 2 * 15, 3* 10, 5 *6

We have the option of using single digits 1,2,3,5 and 6
No : of four digit numbers such that the product is 30
The possible combinations are using digits (1,2,3,5) & (1,1, 5, 6) such that the product 30.
No: of different combinations possible using (1,2,3,5) = 4! = 24
No: of different combinations possible using (5,6,1,1) = 4!/2! = 12 ('1' is repeated twice)
Total 4 digit numbers so that the product is 30 is 24+12 = 36

No: of three digit numbers such that the product is 30
The possible combinations are using digits (2,3, 5) & (1,5,6)
No: of different combinations possible using (2,3,5) = 3! = 6
No: of different combinations possible using (1,5,6) = 3! = 6
Total 3 digit numbers so that the product is 30 is 6+6 = 12

No: of 2 digit numbers such that the product is 30 is 2 (56 and 65)

Total numbers under 10000 such that the product is 30 is 36+12+2 = 50

Ambarish
_________________

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Kindly press Kudos if the explanation is clear.
Thank you
Ambarish

Originally posted by askhere on 20 May 2015, 03:45.
Last edited by askhere on 20 May 2015, 08:14, edited 1 time in total.
Manager
Joined: 21 Feb 2012
Posts: 56
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 07:00
2
My attempt:

We need a number from one digit to four digits whose product is 30. None of the digits can be zero then.
The single digit factors of 30 are 1,2,3,5,6 (rest are more than 1 digit)

So we form the combinations as -

Starting with 4 digit numbers
1 _ _ _ 1st combination of 2,3,5 has 3! and 2nd combination of 5,6,1 has 3!
2 _ _ _ combination of 1,3,5 has 3!
3 _ _ _ combination of 1,2,5 has 3!
5 _ _ _ 1st combination of 1,2,3 has 3! and 2nd combination of 6,1,1 has 3!/2!
6 _ _ _ combination of 5,1,1 has 3!/2!

So total 36 ways

Now 3 digit numbers
156 combinations gives 3! ways
235 combination gives 3! ways

So total 12 ways

Now 2 digit numbers
56 combination gives us 2! ways

So total 2 ways

Grand total of ways = 36+12+2=50

I am kind of confused now as none of the options is 50
_________________

Regards
J

Do consider a Kudos if you find the post useful

Math Expert
Joined: 02 Aug 2009
Posts: 7334
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 07:39
1
30 = 1 * 30, 2 * 15, 3* 10, 5 *6

We have the option of using single digits 1,2,3,5 and 6
The possible combinations are using digits (1,2,3,5) & (1,1, 5, 6) to make their product 30.
No: of different combinations possible using (1,2,3,5) = 4! = 24
No: of different combinations possible using (5,6,1,1) = 4!/2! = 12 ('1' is repeated twice)
Total numbers under 10000 so that the product is 30 = 24+12 = 36

Ambarish

hi,
you are forgetting 3 digits and 2 digits number....
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Math Expert
Joined: 02 Aug 2009
Posts: 7334
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 07:44
1
Bunuel wrote:
How many positive integers less than 10,000 are such that the product of their digits is 30?

A. 12
B. 24
C. 36
D. 38
E. 40

hello,
we can have 2,3 or 4 digits number satisfying the condition..

4 digits number: it can consist of 1,2,3,5 or 1,1,6,5..
1,2,3,5- 4! ways
1,1,5,6- 4!/2 ways
total-36 ways

3 digits number: it can have 2,3,5 or 1,5,6..
2,3,5 - 3! ways
1,5,6 - 3! ways
total - 12 ways

2 digits- 5,6
total - 2 ways

overall- 50 ways...
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Math Expert
Joined: 02 Sep 2009
Posts: 52905
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 07:48
Jackal wrote:
My attempt:

We need a number from one digit to four digits whose product is 30. None of the digits can be zero then.
The single digit factors of 30 are 1,2,3,5,6 (rest are more than 1 digit)

So we form the combinations as -

Starting with 4 digit numbers
1 _ _ _ 1st combination of 2,3,5 has 3! and 2nd combination of 5,6,1 has 3!
2 _ _ _ combination of 1,3,5 has 3!
3 _ _ _ combination of 1,2,5 has 3!
5 _ _ _ 1st combination of 1,2,3 has 3! and 2nd combination of 6,1,1 has 3!/2!
6 _ _ _ combination of 5,1,1 has 3!/2!

So total 36 ways

Now 3 digit numbers
156 combinations gives 3! ways
235 combination gives 3! ways

So total 12 ways

Now 2 digit numbers
56 combination gives us 2! ways

So total 2 ways

Grand total of ways = 36+12+2=50

I am kind of confused now as none of the options is 50

Option E should have been 50. Edited.
_________________
Manager
Joined: 12 Nov 2014
Posts: 63
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 08:16
chetan2u wrote:
30 = 1 * 30, 2 * 15, 3* 10, 5 *6

We have the option of using single digits 1,2,3,5 and 6
The possible combinations are using digits (1,2,3,5) & (1,1, 5, 6) to make their product 30.
No: of different combinations possible using (1,2,3,5) = 4! = 24
No: of different combinations possible using (5,6,1,1) = 4!/2! = 12 ('1' is repeated twice)
Total numbers under 10000 so that the product is 30 = 24+12 = 36

Ambarish

hi,
you are forgetting 3 digits and 2 digits number....

Thanks for pointing out. I edited my answer.
_________________

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Kindly press Kudos if the explanation is clear.
Thank you
Ambarish

Manager
Joined: 18 Nov 2013
Posts: 78
Concentration: General Management, Technology
GMAT 1: 690 Q49 V34
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

20 May 2015, 13:53
1
Q integers >10,000 are such that the product of their digits is 30

possible factors of 30 that can be digits (5,6,1,1) and (2,3,5,1), so two sets to choose from
Set 1 : (2,3,5,1)
Set 2 : (5,6,1,1)

4 Digit numbers _ _ _ _ (4 spots to be filled by 4 digits )

Set 1 : (2,3,5,1) _ _ _ _ 4.3.2.1 = 24
Set 2 : (5,6,1,1) _ _ _ _ (4.3.2.1)/2 = 12

3 Digit numbers _ _ _ (3 spots to be filled by 3 digits )

Set 1 : (2,3,5) _ _ _ 3.2.1 = 6
Set 2 : (5,6,1) _ _ _ 3.2.1 = 6

2 Digit numbers _ _ (2 spots to be filled by 2 digits )

Set 1 : not possible
Set 2 : (5,6) _ _ _ _ 2.1 = 2

total = 36 + 12 + 2 = 50 Ans : E
_________________

_______
- Cheers

+1 kudos if you like

Intern
Status: Online
Joined: 07 Feb 2015
Posts: 28
Location: India
Rudey: RD
Concentration: Marketing, General Management
GMAT 1: 620 Q45 V31
GMAT 2: 640 Q46 V31
GPA: 3.29
WE: Sales (Hospitality and Tourism)
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

21 May 2015, 08:00
chetan2u wrote:
Bunuel wrote:
How many positive integers less than 10,000 are such that the product of their digits is 30?

1,1,5,6- 4!/2 ways
...

Correct me if I'm wrong but you have chosen 1,1,5,6 in 4!/2, is it because integer 1 is being repeated twice in set (1,1,5,6)?
Math Expert
Joined: 02 Aug 2009
Posts: 7334
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

21 May 2015, 18:28
1
RudeyboyZ wrote:
chetan2u wrote:
Bunuel wrote:
How many positive integers less than 10,000 are such that the product of their digits is 30?

1,1,5,6- 4!/2 ways
...

Correct me if I'm wrong but you have chosen 1,1,5,6 in 4!/2, is it because integer 1 is being repeated twice in set (1,1,5,6)?

Hi RudeyboyZ,
you are correct ..
say the set was 1,2,1,1, the ways would have been 4!/3!...here out of 4 positions , there will be 3! will be common .. basically the ways 1 can be placed within themselves...
if the set was 1,1,2,2, the ways would have been 4!/2!2!..
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Non-Human User
Joined: 09 Sep 2013
Posts: 9834
Re: How many positive integers less than 10,000 are such that the product  [#permalink]

### Show Tags

26 Mar 2018, 17:13
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.
_________________
Re: How many positive integers less than 10,000 are such that the product   [#permalink] 26 Mar 2018, 17:13
Display posts from previous: Sort by