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# How many positive integers less than 10,000

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How many positive integers less than 10,000 [#permalink]  16 Oct 2010, 00:42
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Question Stats:

26% (01:37) correct 74% (01:06) wrong based on 74 sessions
How many positive integers less than 10,000 are such that the product of their digits is 210?

24
30
48
54
72

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[Reveal] Spoiler: OA

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Re: How many positive integers less than 10,000 [#permalink]  16 Oct 2010, 00:48
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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 [#permalink]  17 Oct 2010, 05:00
Expert's post
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: integers [#permalink]  28 Oct 2010, 18:24
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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How many positive integers less than 10,000 are such that [#permalink]  29 Jun 2011, 17:37
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: Question [#permalink]  29 Jun 2011, 17:50
1
KUDOS
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: Question [#permalink]  30 Jun 2011, 02:34
1. read question carefully--it says no < 10,000
that means 10,000 < 4 digit > 999and 3 digit < 10002. 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
see gmat will not go complicate these kind of question further so all the best
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Re: Question [#permalink]  30 Jun 2011, 02:44
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 9999 are such that [#permalink]  10 Feb 2013, 13:05
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 [#permalink]  29 Sep 2013, 08:26
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Re: How many positive integers less than 10,000 are such that   [#permalink] 29 Sep 2013, 08:26
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