BTGmoderatorDC wrote: ↑Mon May 25, 2020 6:39 pm
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
OA
C
Solution:
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
