How many five-digit prime numbers can be obtained by using all the digits 1, 2, 3, 4 and 5 without repetition of digits?
Answer & explanation
Answer: (a) Zero
Every such number uses the digits 1 to 5 once, so its digit sum is 15. A digit sum of 15 makes every one of these numbers divisible by 3, so none can be prime.
- Digit sum = 1 + 2 + 3 + 4 + 5 = 15, whatever the order.
- 15 is divisible by 3, so every such five-digit number is divisible by 3.
- A number greater than 3 that is divisible by 3 is not prime, so the count is zero.
Remember · Before listing permutations for primes, check the digit sum: if it is a multiple of 3, no arrangement is prime.
Question and answer: UPSC's official GS Paper II (2020, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·