How many positive integers less than 10,000

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How many positive integers less than 10,000 [#permalink]

16 Oct 2010, 01:42

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12% (01:22) correct

87% (01:01) wrong

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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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Re: How many positive integers less than 10,000 [#permalink]

16 Oct 2010, 01:48

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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!

Answer = 24+24+6 = 54

Answer is (d)
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Re: How many positive integers less than 10,000 [#permalink]

17 Oct 2010, 06:00

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

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Re: integers [#permalink]

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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how many positive integers less than 9999 are such that [#permalink]

10 Feb 2013, 12:20

how many positive integers less than 9999 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 9999 are such that [#permalink]

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 9999 are such that [#permalink]

10 Feb 2013, 20:47

yes, u r right...it's 54.

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Re: how many positive integers less than 9999 are such that [#permalink] 10 Feb 2013, 20:47

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