Three positive integers a, b, c with a>c satisfy the following equations: a c+b+c=b c+a+66, a+b+c=32 . Find the value of a .

Step-by-step solution

aligned & a>c & A, b, c l⁺ & a c+b+c=b c+a+66, a+b+c=32 & c(a-b)+(b-a)+c=66 & (a-b)(c-1)+(c-1)=66-1 & (c-1)(a-b+1)=65=1 65=65 1 & =5 13=13 5 aligned Case-(i): c-1=65, a-b+1=1 c=66 , but a+b+c=32 not possible Case(ii): c-1=1, a-b+1=65c=2, a-b=64a+b+c=32 a+b=30 2 a=64 a=47, b=-17 not possible Case(iii): c-1=5, a-b+1=13 aligned & c=6, a+ i =26 & a-b=12 & a=19, b=7 & Case(iv): & c-1=13, a-b+1=5 & c-1=13, a-b+1=5 & c=14, a-b=4 & a=b=18 & a=11 but a < c & only a=19 aligned