CAT Exam Model Paper 8 with solutions for free online practice

Case (1) - n is odd: Then 8${}^{\mathit{n}}$ + 2${}^{\mathit{n}}$ is divisible by (8 + 2). Similarly 7${}^{\mathit{n}}$ + 3n${}^{\mathit{n}}$ is divisible by (7 + 3). So the remainder when the expression is divided by 10 is 0.
Case (2) - n is even. Then 8n${}^{\mathit{n}}$+ 2${}^{\mathit{n}}$ is not divisible by (8 + 2). Similarly 7${}^{\mathit{n}}$ + 3${}^{\mathit{n}}$ is not divisible by (7 + 3). So we shall find the last digits to find the remainders.
n is an odd multiple of 2: The last digits of 8${}^{\mathit{n}}$, 7${}^{\mathit{n}}$, 3${}^{\mathit{n}}$, 2n${}^{\mathit{n}}$ are 4,9,9,4 respectively. So the remainder when the expression is divided by 10 is 6.
n is an even multiple of 2: The last digits of 8${}^{\mathit{n}}$, 7${}^{\mathit{n}}$, 3${}^{\mathit{n}}$, 2${}^{\mathit{n}}$ are 6,1,1,6 respectively. So the remainder when the expression is divided by 10 is 4.
So the sum of all the possible distinct remainders is 0 + 6 + 4 = 10