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28 Jun 2021, 04:15
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A
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Difficulty:
35%
(medium)
Question Stats:
69% (01:33) correct
31%
(01:51)
wrong
based on 106
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Of all the five digit numbers which can be formed using digits 1, 2, 3, 4 and 5, how many are prime numbers (repetition of digits is not allowed)?
(A) 0
(B) 1
(C) 4
(D) 8
(E) 12
(A) 0
(B) 1
(C) 4
(D) 8
(E) 12
28 Jun 2021, 04:23
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Bunuel
The sum of the digits of any number formed using (1, 2, 3, 4 and 5) will always be 15.
15 = 5*3
Hence, no matter what number is formed, it will always be divisible by 3... which means the number won't be a prime number.
Option A.
28 Jun 2021, 05:03
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The only way to definitively prove a number like 12,347 is prime (and it is) is to try dividing it by every single prime number less than √12347, which is roughly 111. If you can't divide it by a single prime less than 111, it is a prime number. So you'd need to try dividing by 29 different numbers, most of which are very awkward to divide by -- numbers like 79 and 97. To prove even one number this size is prime would take half an hour. So if the answer to a question like this was one or greater, the question would take hours to solve. The only conceivable answer here is zero -- there must be some easy way to prove none of these numbers is prime. And we can often quickly prove large numbers are not prime. Large even numbers aren't prime, and large numbers ending in '5' are not prime, and large numbers with a digit sum that is a multiple of 3 are not prime. As ProKiller points out, that's how we can see none of these numbers is prime -- the digits sum to 15 -- but you could confidently pick 'zero' here without even checking that.
