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.
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How many positive integers less than 2*10^4 are there in
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BTGmoderatorLU
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Brent@GMATPrepNow
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In other words, "How many positive integers less than 20,000 are there in which each digit is a prime number?"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.
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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deloitte247
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$$2\cdot10^4=20000$$
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
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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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.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
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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