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
OA D
Source: Magoosh
How many positive integers less than 10,000 are such that the product of their digits is 210?
This topic has expert replies

 Moderator
 Posts: 6253
 Joined: 07 Sep 2017
 Followed by:20 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
\(210 = 2\times5\times3\times7 = 5\times6\times7\times1 = 5\times6\times7\)BTGmoderatorDC wrote: ↑Tue Nov 23, 2021 6:54 pmHow 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
OA D
Source: Magoosh
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 \(5,6,7\) (3digit numbers) \(= 3!\)
Answer \(= 24+24+6 = 54\)
GMAT/MBA Expert
 [email protected]
 GMAT Instructor
 Posts: 15955
 Joined: 08 Dec 2008
 Location: Vancouver, BC
 Thanked: 5254 times
 Followed by:1267 members
 GMAT Score:770
210 = (2)(3)(5)(7)BTGmoderatorDC wrote: ↑Tue Nov 23, 2021 6:54 pmHow 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
OA D
Source: Magoosh
We need to consider 3 cases:
Case 1: 4digit 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: 4digit numbers using 1, 6, 5, 7
There are 4 digits, so this can be accomplished in 4! (24) ways
Case 3: 3digit 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