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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

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 adding all the case 1,2,3= 54that is our answer D. see gmat will not go complicate these kind of question further so all the best _________________

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? _________________

Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.

Re: How many positive integers less than 10,000 are such that [#permalink]
29 Sep 2013, 08:26

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