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How many positive integers less than 2*10^4 are there in
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29 Oct 2017, 01:14
29 Oct 2017, 01:14
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Question Stats:
44% (02:06) correct
55% (02:11) wrong based on 58 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
Kudos for correct solution.
(A) 256
(B) 326
(C) 340
(D) 625
(E) 775
Kudos for correct solution.
Re: How many positive integers less than 2*10^4 are there in
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30 Oct 2017, 08:59
30 Oct 2017, 08:59
3
The upper bound of our set is 2*10^4 = 20,000. Thus, the highest number in the set is 19,999. The problem is that for a 5-digit number to be less than 20,000, the ten-hundredths digit must be 1 and 1 is not a prime number. Thus, we must move down our boundary to 9,999, 4-digit number.
Then, the prime numbers made of one digit are 2, 3, 5, 7. Thus they are 4. Thus, 4-digit numbers made of prime number digits are 4*4*4*4 = 256 because each digit can assume four values and they can be repeated. Then, 3-digit numbers are 4*4*4 = 64, 2-digit numbers are 4*4 = 16 and 1-digit number are 4.
Summing up, we get 256+64+16+4 = 340. Answer C
Then, the prime numbers made of one digit are 2, 3, 5, 7. Thus they are 4. Thus, 4-digit numbers made of prime number digits are 4*4*4*4 = 256 because each digit can assume four values and they can be repeated. Then, 3-digit numbers are 4*4*4 = 64, 2-digit numbers are 4*4 = 16 and 1-digit number are 4.
Summing up, we get 256+64+16+4 = 340. Answer C
Re: How many positive integers less than 2*10^4 are there in
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04 Nov 2023, 02:17
04 Nov 2023, 02:17
1
x<20,000
Since we need all digits to be prime, we can exclude the case of 5 digit numbers since they can only start with 1, and 1 is not prime.
Now lets use box method:
There are 4 prime digits - 2,3,5, and 7. Therefore for a
4 digit number you can have: 4*4*4*4=4^4
3 digit: 4*4*4=4^3
2 digit: 4*4=4^2
1 digit: 4
Total: 256 + 64 + 16 + 4 which sums to 340.
C.
Since we need all digits to be prime, we can exclude the case of 5 digit numbers since they can only start with 1, and 1 is not prime.
Now lets use box method:
There are 4 prime digits - 2,3,5, and 7. Therefore for a
4 digit number you can have: 4*4*4*4=4^4
3 digit: 4*4*4=4^3
2 digit: 4*4=4^2
1 digit: 4
Total: 256 + 64 + 16 + 4 which sums to 340.
C.
