f i rs t e q u a t i o n : pq + p r = 80 pq = 80 − p r .......... [ 1 ] seco n e q u a t i o n : pq + q r = 425.......... [ 2 ]
N o w s u b s t i t u t in g e q u a t i o n 1 in t o 2 : pq + q r = 425 ( 80 − p r ) + q r = 425 − p r + q r = 425 − 80 − p r + q r = 345 r ( q − p ) = 345 w e ha v e : p ( q + r ) = 80 = 2 4 × 5 , q ( p + r ) = 425 = 5 2 × 17 r ( q − p ) = 345 = 3 × 5 × 23 so : p = 2 or 5 , q = 5 or 17 , r = 3 , 5 , or 23.
F ro m : p ( q + r ) = 80 , i f p = 5 t h e n q + r = 16 , w hi c h ha s n o so l u t i o n S o p = 2 , t h e n q + r = 40 , h e n ce q = 17 an d r = 23. ∴ p + q + r i s 2 + 17 + 23 = 42
We found the prime numbers satisfying the given equations are p = 2 , q = 17 , and r = 23 . The sum of these numbers is 42 . Therefore, p + q + r = 42 .
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