Question

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

• A. 1
• B. 4
• C. 6
• D. 8

Answer 1

The Correct Option is D

To find the remainder when \(11^{1011} + 1011^{11}\) is divided by 9, we can use various properties of modular arithmetic to simplify the problem.

Step 1: Simplify \(11^{1011} \mod 9\)

First, note that \(11 \equiv 2 \mod 9\). Therefore, we have:

\(11^{1011} \equiv 2^{1011} \mod 9\)

Using Euler’s theorem, \(a^{\phi(n)} \equiv 1 \mod n\) where \(gcd(a, n) = 1\) and \(\phi(n)\) is the Euler's totient function. For \(n = 9\), \(\phi(9) = 6\).

So, \(2^6 \equiv 1 \mod 9\).

Now, reduce the exponent 1011 mod 6:

\(1011 \equiv 3 \mod 6\)

Thus, \(2^{1011} \equiv 2^3 \mod 9\).

Calculate \(2^3\):

\(2^3 = 8\)

So, \(2^{1011} \equiv 8 \mod 9\).

Step 2: Simplify \(1011^{11} \mod 9\)

Next, note that \(1011 \equiv 3 \mod 9\). Therefore:

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

As calculated earlier, anything of the form \(3^n \mod 9\) where \(n \geq 2\) is 0 because \(3^2 = 9 \equiv 0 \mod 9\). Thus:

\(3^{11} \equiv 0 \mod 9\)

Step 3: Combine Results

Finally, add the components together with modulo 9:

\((11^{1011} + 1011^{11}) \mod 9 \equiv (8 + 0) \mod 9\)

This results in:

\(8 \mod 9 = 8\)

Conclusion

The remainder when \(11^{1011} + 1011^{11}\) is divided by 9 is 8.