Case (1) - n is odd: Then 8‌n + 2‌n is divisible by (8 + 2). Similarly 7‌n + 3n‌n is divisible by (7 + 3). So the remainder when the expression is divided by 10 is 0. Case (2) - n is even. Then 8n‌n+ 2‌n is not divisible by (8 + 2). Similarly 7‌n + 3‌n 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 8‌n, 7‌n, 3‌n, 2n‌n 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 8‌n, 7‌n, 3‌n, 2‌n 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