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

Question

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

Bunuel · Accepted Answer

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.

Answer: D.

shrouded1 · Answer

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)

IanStewart · Answer

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?

BrentGMATPrepNow · Answer

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

So, the answer is D

Cheers, 
Brent

EMPOWERgmatRichC · Answer

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

Final Answer:  D

GMAT assassins aren't born, they're made, 
Rich

ScottTargetTestPrep · Answer

Solution:
 
Note that the prime factorization of 210 = 2 x 105 = 2 x 3 x 5 x 7.

Since no two-digit numbers multiply to 210, the integer must be either 3 digits or 4 digits (if it’s less than 10,000).

If it’s a 3-digit integer, then the 3 digits must be {5, 6, 7} (notice that 5 x 6 x 7 = 210). Since there are 3! = 6 ways to permute the 3 digits, there are six 3-digit integers such that the product of their digits is 210.

If it’s a 4-digit integer, then the 4 digits must be {2, 3, 5, 7} (notice that 2 x 3 x 5 x 7 = 210) or {1, 5, 6, 7} (notice that 1 x 5 x 6 x 7 = 210). Since there are 4! = 24 ways to permute the 4 digits, there are 24 four-digit integers in each group such that the product of their digits is 210. In total, there are 24 + 24 = 48 four-digit integers such that the product of their digits is 210.

Therefore, there are 6 + 48 = 54 such integers.

Answer: D

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

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