Positive integers a, b, c satisfy a b a-b =c . What is the largest possible value of a+b+c not exceeding 99?

Step-by-step solution

aligned & 1 b = 1 a + 1 c & b < (a, c) aligned Case I: When a=c , then b= a 2 b even then, a+b+c=5 b (a+b+c)_ =95 Case II: When a c Let a= a₁ and c= c₁ where gcd (a₁, c₁ )=1 So, 1 b = a₁+c₁ a₁ c₁ b= ( a₁+c₁ ) a₁ c₁ So, we must select such that (a₁+c₁ ) If we check for maximum value of (a+b+c)=99 then (a₁+c₁ ) must be a divisor of 99 . It is possible when (a₁, c₁ )=(1,2) , then is divisible by 3 . (say =3 ) aligned & a=3 , c=6 and b=2 & a+b+c=11 aligned If we take =9,(a+b+c)_ =99 .