Medium Hard Difficulty Level: How many positive integers...

[hansoo]

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

[Stuart@KaplanGMAT]

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

Let's start by breaking 210 down into primes:

210 = 10 * 21 = 2*5*3*7

So, to start, we know that all permutation of 2,3,5,7 will fulfil the requirements: 4! = 4*3*2 = 24

However, we have to recognize that we may be able to use 1 and another number in place of a pair of our numbers.

Since each digit has to be less than or equal to 9, the only substitution we can do is to use 1*6 instead of 2*3 (2*5, the next smallest pair of primes, would give us 10).

So, we can also use the digits 1,5,6,7. Same as last time, this gives us 4! = 24 possible arrangements.

So, the final answer is 24+24=48... choose C

[Stuart@KaplanGMAT]

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

Let's start by breaking 210 down into primes:

210 = 10 * 21 = 2*5*3*7

So, to start, we know that all permutation of 2,3,5,7 will fulfil the requirements: 4! = 4*3*2 = 24

However, we have to recognize that we may be able to use 1 and another number in place of a pair of our numbers.

Since each digit has to be less than or equal to 9, the only substitution we can do is to use 1*6 instead of 2*3 (2*5, the next smallest pair of primes, would give us 10).

So, we can also use the digits 1,5,6,7. Same as last time, this gives us 4! = 24 possible arrangements.

So, the final answer is 24+24=48... choose C

Actually, I made a mistake! (I know, hard to conceive!)

Nowhere does it say we need 4 digits. So, we can also make some 3 digit numbers using 5, 6 and 7.

That's 3! more possibilities.. add another 6 to our 48... choose D!

[hansoo]

Good catch Stuart! This question is a little tricky. The official answer is D.

Here's a quick summary of the explanation:


Here's a video explanation: GMAT video

[co_lor]

Why cant you include
15*5*7 = 3!
125*7=2
so on?
15 is an integer less than 10,000 right? Yes its not a single digit but its only mentioned "positive integers less than 10,000"

Thanks

[winniethepooh]

Hey color, none of your queries will give a product of 210!

[akshayasalunke]

@co_lor: The Q says, product of their "digits" (0-9)