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IPMAT 2025 Exam Solved Papers

Question : 40 of 90

Marks: +1, -0

11^{1011}+1011^{11}

is divided by 9 is

Solution:

11 when divided by 9 remainder is 2

1011 when divided by 9 remainder is 3

Now,

11 \equiv 2 \bmod (9)

so,

11^{1011} \equiv 2^{1011} \bmod (9) \longrightarrow(1)

Now

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

Also,

8 \equiv-1 \bmod (9)

So,

8^{337} \equiv-1 \bmod (9)

So, we can say

2^{1011} \equiv 8^{337} \bmod (9) \equiv-1 \bmod (9)

So,

11^{1011} \equiv-1 \bmod (9)

(From (1))

Now, -1 is a negative remainder, so positive remainder will be

8 \longrightarrow \ldots > (2)

Similarly,

1011 \equiv 3 \bmod (9)

So,

1011^{11} \equiv 3^{11} \bmod (9)

Now,

3^{11}=3^{2} \cdot 3^{9}=9 \cdot 3^{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

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