Question

If x, y, z and w are natural numbers, then what is the sum of all the possible distinct remainders when $9^x+6^y+4^z+{11}^z$ divided by 10? ___

Answer 1

There are four possible scenarios for the units' digit combination of the four numbers. These essentially arise from the even/odd combinations taking x and z:
Scenario I: Both even: $1+6+6+1\to$ Remainder = 4;
Scenario 2: Both odd: $9+6+4+1\to$ Remainder =0
Scenario 3: x odd, z even: $9+6+6+1\to$ Remainder =2
Scenario 4: x even, y odd: $1+6+4+1\to$ Remainder =2
Thus, the sum of all distinct remainders = 4 + 0 + 2 = 6