What is the remainder when $((11^{12})^{13})^{14}$ is divided by 9?

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Let us expand the term

((11^{12})^{13})^{14}

((11^{12})^{13})^{14} = ((9+2)^{12})^{13})^{14}

When we expand the term using binomial expansion, all terms except the last will be divisible by 9.

((2^{12})^{13})^{14} = ((2^3)^4)^{13})^{14} = ((8)^4)^{13})^{14} = ((9-1)^4)^{13})^{14}

Again only last term will not be divisible by 9, when the term is expanded.

((-1)^4)^{13})^{14} = (1^{13})^{14} = 1

So remainder = 1.…

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What this question tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Forgetting that 1 is not prime and that every integer divides 0.