The remainder when $$11^{1011} + 1011^{11}$$ is divided by 9 is

11 when divided by 9 remainder is 2

1011 when divided by 9 remainder is 3

Now, $$11≡2\ mod(9)$$

so, $$11^{1011}≡2^{1011\ }mod(9)$$ ----->(1)

Now $$2^{1011}=\left(2^3\right)^{337}=8^{337}$$

Also, $$8≡-1mod(9)$$

So, $$8^{337}≡-1mod(9)$$

So, we can say $$2^{1011}≡8^{337}mod(9)≡-1\ mod(9)$$

So, $$11^{1011}≡-1\ mod(9)$$ (From (1))

Now, $$-1$$ is a negative remainder, so positive remainder will be 8 --------->(2)

Similarly, $$1011≡3\ mod(9)$$

So, $$1011^{11}≡3^{11\ }mod(9)$$

Now, $$3^{11}=3^2\cdot3^9=9\cdot3^9$$ ,so it is definitely divisible by 9

So, in this part remainder will be $$0$$ --------->(3)

From (2) and (3) we can say, remainder=$$8+0=8$$