How many positive integers less than 2*10^4 are there in which each di

Question

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

quantumliner · Accepted Answer

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

Answer is C

Bunuel · Answer

2. Properties of Integers  
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

Luckisnoexcuse · Answer

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

TheMechanic · Answer

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.

Nikkb · Answer

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

Answer: C

manishgmat1 · Answer

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

AkshatN · Answer

_ - 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

BrentGMATPrepNow · Answer

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

KanishkM · Answer

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

dracarys007 · Answer

1 digit 4
2 digit 4*4 = 16
3 digit 4*4*4 = 64
4 digit 4*4*4*4 =256
Total : 4+16+64+256 = 340

Ambanis · Answer

Any number having unit digit 5 is divisible by 5 then how is it prime number?

Bunuel · Answer

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

The digits of a number should be primes, not the number itself.

bumpbot · Answer

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