The number of triplets a , b , c of positive integers satisfying the equation a 3 + 1 a 2 b a 2 c a b 2 b 3 + 1 b 2 c a c 2 b c 2 c 3 + 1 = 30 is equal to

Step-by-step solution

a b c a b c a 3 + 1 a 2 b a 2 c a b 2 b 3 + 1 b 2 c a c 2 b c 2 c 3 + 1 = 30 ⇒ a b c a 2 + 1 a a 2 a 2 b 2 b 2 + 1 b b 2 c 2 c 2 c 2 + 1 c = 30 ⇒ a 3 + 1 a 3 a 3 b 3 b 3 + 1 b 3 c 3 c 3 c 3 + 1 = 30 Apply R 1 ↔ R 1 + R 2 + R 3 ⇒ a 3 + b 3 + c 3 + 1 1 1 1 b 3 b 3 + 1 b 3 c 3 c 3 c 3 + 1 = 30 Apply C 2 ↔ C 2 - C 1 ,   C 3 ↔ C 3 - C 1 ⇒ a 3 + b 3 + c 3 + 1 1 0 0 b 3 1 0 c 3 0 1 = 30 ⇒ a 3 + b 3 + c 3 + 1 = 30 ⇒ a 3 + b 3 + c 3 = 29 a , b , c ≡ 1,1 , 3 or