How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?
- (a) Zero ✓ UPSC's answer
- (b) One
- (c) Nine
- (d) Ten
Why the answer is (a)
• A number is divisible by 5 if its last digit is 0 or 5.
• The set of available digits is {1, 2, 3, 4, 5}, which does not contain 0.
• Therefore, any five-digit number formed using all these digits without repetition must end with the digit 5.
• Since every such number ends in 5, it is divisible by 5.
• A prime number greater than 5 cannot be divisible by 5.
• Consequently, no such five-digit number can be prime, resulting in a count of zero.
Why the other options are wrong
- (b) One
- It is impossible to form a prime number because all permutations end in 5 and are thus divisible by 5.
- (c) Nine
- The count is not nine because the divisibility rule for 5 applies to all 120 possible permutations, making them all composite.
- (d) Ten
- The count is not ten because no permutation of these digits yields a prime number due to the mandatory trailing 5.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.