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

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by Brent@GMATPrepNow » Sat Sep 01, 2018 5:27 am
BTGmoderatorLU wrote:Source: Magoosh

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

The OA is C.
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

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by deloitte247 » Sun Sep 02, 2018 10:46 am
$$2\cdot10^4=20000$$
Positive integers less than 20,000 starts from 19999 in a descending order with at most 5 digits with 1 at the first place and we cannot have a prime number for the first place because it will make the value to be more than 20,000. So, we start with 2nd place.
We can have only 2,3,5 and 7 as single digit prime numbers filling in the remaining 4 place.
For a 4-digit number, there are 4 options for each digit.
Therefore, 4*4*4*4=256 possible numbers

for a 3-digit number, there are 3 options for each digit. Thus, 4*4*4=64 possible numbers

For a 2-digit number, there are 2 options for each digit. Thus, 4*4=16 possible numbers

For 1-digit number, there is just the option for 4 possible numbers (2,3,5 and 7).

Total possible number = 256+64+16+4 =340

Therefore, option C is the correct answer
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by Scott@TargetTestPrep » Tue Sep 11, 2018 4:16 pm
BTGmoderatorLU wrote:Source: Magoosh

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
We note that 2*10^4 = 2 * 10,000 = 20,000. We need to determine the number of integers less than 20,000 in which each digit is a prime number. The prime single-digit numbers are 2, 3, 5, and 7. Since 1 is not a prime, we see that we can rule out all numbers greater than or equal to 10,000. In other words, the number must be no more than 4 digits.

If it's a 4-digit number, then there are 4 x 4 x 4 x 4 = 4^4 = 256 such numbers.

If it's a 3-digit number, then there are 4 x 4 x 4 = 4^3 = 64 such numbers.

If it's a 2-digit number, then there are 4 x 4 = 4^2 = 16 such numbers.

If it's a 1-digit number, then there are 4 such numbers.

So there are 256 + 64 + 16 + 4 = 340 such numbers.

Answer: C

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