Three prime numbers $p$, $q$ and $r$, each less than 20, are such that $p - q = q - r$. How many distinct…
Three prime numbers $p$, $q$ and $r$, each less than 20, are such that $p - q = q - r$. How many distinct possible values can we get for $(p + q + r)$?
4
5
6
More than 6
Solution
Primes less than 20: 2, 3, 5, 7, 11, 13, 17, 19. The condition $p-q = q-r$ means $p, q, r$ form an arithmetic progression, so $p + q + r = 3q$. Three-term AP triples of primes within range: (3,5,7) sum 15, (3,7,11) sum 21, (5,11,17) sum 33, (3,11,19) sum 33, (7,13,19) sum 39. Distinct sums obtained: 15, 21, 33, 39 - giving 4 distinct values of $(p+q+r)$.