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

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

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16 Oct 2010, 01:42
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21% (01:49) correct 79% (01:56) wrong based on 315 sessions

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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

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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17 Oct 2010, 06:00
4
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.

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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16 Oct 2010, 01:48
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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!

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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28 Oct 2010, 19:24
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...
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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29 Jun 2011, 18:50
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"
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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30 Jun 2011, 03:34
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
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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30 Jun 2011, 03:44
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.
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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10 Feb 2013, 14:05
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?
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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06 Sep 2014, 22:59
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
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how many positive integer less than 10,000 such that product of their  [#permalink]

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05 Mar 2015, 06:01
how many positive integer less than 10,000 such that product of their digits is 210?
ans- 54
how come??
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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05 Mar 2015, 06:05
anik1989 wrote:
how many positive integer less than 10,000 such that product of their digits is 210?
ans- 54
how come??

Merging topics.

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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05 Mar 2015, 21:58
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
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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06 Mar 2015, 01:47
I love your sharing,i learn a lot from your post.There is a lot of very useful knowledge in your post.
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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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20 Jul 2016, 06:14
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.

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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20 Jul 2016, 06:35
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

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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20 Jul 2016, 06:49
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.

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Re: How many positive integers less than 10,000 are such that the product  [#permalink]

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29 Mar 2018, 11:58
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

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Re: How many positive integers less than 10,000 are such that the product &nbs [#permalink] 29 Mar 2018, 11:58
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