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Math Expert V
Joined: 02 Sep 2009
Posts: 55609
How many positive integers less than 2 x 10^4 are there in which each  [#permalink]

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15 00:00

Difficulty:   95% (hard)

Question Stats: 53% (02:34) correct 47% (02:25) wrong based on 189 sessions

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How many positive integers less than 2*10^4 are there in which each digit is a prime number?

(A) 256
(B) 326
(C) 340
(D) 625
(E) 775

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Senior Manager  G
Joined: 24 Apr 2016
Posts: 328
Re: How many positive integers less than 2 x 10^4 are there in which each  [#permalink]

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4
4
The positive integers less than 20000, can be 1 digit, 2 digit, 3 digit, 4 digit and 5 digit.

Now each digit in the above integers must be a prime number.

1 Digit : Possible numbers - 2, 3, 5, 7 : Total = 4

2 Digits: Both the hundreds and Units digit can take any value from 2, 3, 5, 7: So Total : 4 * 4 = 16

3 Digits: The Thousands, hundreds and Units digit can take any value from 2, 3, 5, 7: So Total : 4 * 4 * 4 = 64

4 Digits: The Ten Thousands, Thousands, hundreds and Units digit can take any value from 2, 3, 5, 7: So Total : 4 * 4 * 4 * 4 = 256

5 Digits: As the least possible 5 digit number has to to start with 2, which will make the 5 digit number greater than 20000, hence there is 0, 5 digit numbers less than 20000, with each digit as a prime number. Total = 0

Therefore adding all : 4 + 16 + 64 + 256 = 340

General Discussion
Current Student P
Joined: 18 Aug 2016
Posts: 618
Concentration: Strategy, Technology
GMAT 1: 630 Q47 V29 GMAT 2: 740 Q51 V38 Re: How many positive integers less than 2 x 10^4 are there in which each  [#permalink]

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Bunuel wrote:
How many positive integers less than 2*10^4 are there in which each digit is a prime number?

(A) 256
(B) 326
(C) 340
(D) 625
(E) 775

20000 (5 digits)
No number can be formed of 5 digits as 1 is not a prime number and at First place only Prime number 2 can come

single digit prime numbers are
2,3,5,7
4 digit numbers with prime numbers will be
So 4^4 = 256
3 digit numbers with prime numbers will be
4^3 = 64
2 digit numbers with prime numbers will be
4^2 = 16
1 digit numbers with prime numbers will be
4
Total = 256 + 64 + 16 + 4 = 340

C
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Luckisnoexcuse
Senior Manager  G
Joined: 19 Oct 2012
Posts: 306
Location: India
Concentration: General Management, Operations
GMAT 1: 660 Q47 V35 GMAT 2: 710 Q50 V38 GPA: 3.81
WE: Information Technology (Computer Software)
How many positive integers less than 2 x 10^4 are there in which each  [#permalink]

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1
Numbers less than 20,000 will have 5 digits or less.
This is a generic representation of this number: _ _ _ _ _
Clearly, we can't have a prime number for the 1st place since that would take the value more than 20k. Therefore, we start with 2nd place and so on. We can have only 2,3,5,7 as single digit prime numbers filling in the places.
So for a 4 digits number: we have 4 options for each digit. This gives us $$4 . 4 . 4 . 4 = 256$$ possible numbers.
And for a 3 digits number: we have 4 options for each digit. This gives us $$4 . 4 . 4 = 64$$ possible numbers.
And for a 2 digits number: we have 4 options for each digit. This gives us $$4 . 4 = 16$$ possible numbers.
And for a 1 digit number: we have 4 options for the digit. This gives us $$4 . 1 = 4$$ possible numbers.
Adding all possible numbers, we get 340. Hence C.
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Math Expert V
Joined: 02 Sep 2009
Posts: 55609
How many positive integers less than 2*10^4 are there in which each di  [#permalink]

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Current Student P
Joined: 02 Jul 2017
Posts: 294
Concentration: Entrepreneurship, Technology
GMAT 1: 730 Q50 V38 How many positive integers less than 2*10^4 are there in which each di  [#permalink]

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1
1
Integers less than 20000 in which each digit is a prime number : 2,3,5,7,22,23,25,27,32,33,35,37......

possible options
Single digit -> 4
2 digit -> 4 *4 = 16
3 digit -> 4*4*4 = 64
4 digit -> 4*4*4*4 =256
5 digit :
10000 till 20000 -> But here as 1st digit is always 1 which is not prime.. we cannot consider numbers between 10000 and 20000

Total : 4+16+64+256 = 340

Originally posted by Nikkb on 29 Oct 2017, 02:32.
Last edited by Nikkb on 29 Oct 2017, 03:09, edited 2 times in total.
Intern  B
Joined: 11 Mar 2014
Posts: 23
Schools: HEC Montreal '20
How many positive integers less than 2*10^4 are there in which each di  [#permalink]

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is this the correct method
2* 10^4=20,000
single digit 2,3,5,7
two digit = 4*4 ie 16
three digit numbers 4*4*4= 64
four digit =4*4*4*4=25
6
we have reached 10,000
as tenth thousnadths digit is 1 all numbers beyond will have this number(1x,xxx) as non prime which makes them null and void
also the range 20,000 forbids us to consider 22,222
so we have to consider
256+64+16+4=340
Am i correct
Intern  B
Joined: 13 Feb 2017
Posts: 5
Re: How many positive integers less than 2*10^4 are there in which each di  [#permalink]

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_ - 4 ways
_ _ - 4 x 4 = 16 ways
_ _ _ - 4x4x4 = 64 ways
_ _ _ _ - 4x4x4x4= 256ways total = 340

Sent from my SM-G615F using GMAT Club Forum mobile app
CEO  V
Joined: 12 Sep 2015
Posts: 3777
Re: How many positive integers less than 2*10^4 are there in which each di  [#permalink]

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Top Contributor
Bunuel wrote:
How many positive integers less than 2*10^4 are there in which each digit is a prime number?

(A) 256
(B) 326
(C) 340
(D) 625
(E) 775

In other words, "How many positive integers less than 20,000 are there in which each digit is a prime number?"
The prime digits are: 2, 3, 5 and 7
Notice that, using the digits 2, 3, 5 and 7, we cannot create a 5-digit number that's less than 20,000
So, we must consider 4 possible cases: 4-digit numbers, 3-digit numbers, 2-digit numbers, and 1-digit numbers

4-digit numbers
There are 4 options for the first digit (2, 3, 5 or 7), 4 options for the second digit, 4 options for the third digit, and 4 options for the last digit
TOTAL number of 4-digit numbers = (4)(4)(4)(4) = 256

3-digit numbers
There are 4 options for the first digit (2, 3, 5 or 7), 4 options for the second digit, and 4 options for the last digit
TOTAL number of 3-digit numbers = (4)(4)(4) = 64

2-digit numbers
There are 4 options for the first digit (2, 3, 5 or 7), and 4 options for the last digit
TOTAL number of 2-digit numbers = (4)(4) = 16

1-digit numbers
There are 4 options: 2, 3, 5, 7

ANSWER = 256 + 64 + 16 + 4 = 340 = C

Cheers,
Brent
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VP  G
Joined: 09 Mar 2018
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Re: How many positive integers less than 2*10^4 are there in which each di  [#permalink]

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Bunuel wrote:
How many positive integers less than 2*10^4 are there in which each digit is a prime number?

(A) 256
(B) 326
(C) 340
(D) 625
(E) 775

Numbers less than 20000, in which each digit is a prime number

There will be 4 prime numbers 2 3 5 7

_ _ _ _ => 4*4*4*4 = 216
_ _ _ => 4*4*4 = 64
_ _ => 4*4= 16
_=> 4

216 + 64 + 20
340

C
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Many of life's failures happen with people who do not realize how close they were to success when they gave up. Re: How many positive integers less than 2*10^4 are there in which each di   [#permalink] 01 Feb 2019, 03:51
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