Three prime numbers p, q and r, each less than 20, are such that p − q = q − r. How many distinct…

Answer: (d) More than 6

Why not the tempting option · UPSC's key is (d). The item says 'three prime numbers p, q and r', not three different primes, and uses 'distinct' only for the sums — so p = q = r is allowed, every prime below 20 can be the middle term and there are 8 sums. Assuming the primes must differ gives 4 sums, option (a), but that assumption is not in the item. In the exam, do not add 'distinct' where the question has not written it.

Since p − q = q − r, the three primes are in arithmetic progression and p + q + r = 3q. The item does not say the primes must be different, so p = q = r is allowed and each of the 8 primes below 20 can be q, giving 8 distinct sums — more than 6.

1. p − q = q − r gives p + r = 2q, so p + q + r = 3q. The sum depends only on the middle prime q.
2. The item does not require the primes to be different, so p = q = r is allowed (both differences are then 0).
3. So every prime q below 20 gives a value: q = 2, 3, 5, 7, 11, 13, 17, 19 gives p + q + r = 6, 9, 15, 21, 33, 39, 51, 57.
4. That is 8 distinct values, which is more than 6.
5. Trap: if one assumed the primes must be different, only (3, 5, 7), (3, 7, 11), (3, 11, 19), (5, 11, 17) and (7, 13, 19) would work, giving 4 sums — but the item says 'distinct' only of the sums, not of the primes, so equal primes count.

Remember · Equal differences mean an arithmetic progression: the sum of three terms is three times the middle one. Apply 'distinct' only where the question states it.

Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 1 Oct 2026 (how we verify). · Report an error