GATE CS 2016 Set 2 Q.39 (CS Q.29): Combinatorics, Counting, Recurrence Relations and Generating Functions

The value of the expression $13^{99} \pmod{17}$ , in the range 0 to 16, is ________.

Your answer

Official answer and solution

Solution

Apply Fermat's Little Theorem to reduce the power modulo the prime 17.

We want to evaluate $13^{99} \pmod{17}$ . Since 17 is a prime number and $\gcd(13, 17) = 1$ , by Fermat's Little Theorem:

13^{17-1} = 13^{16} \equiv 1 \pmod{17}

Express the exponent 99 in terms of multiples of 16:

99 = 16 \times 6 + 3

Using the properties of modular arithmetic:

13^{99} = (13^{16})^6 \times 13^3 \equiv (1)^6 \times 13^3 \equiv 13^3 \pmod{17}

Compute $13^3 \pmod{17}$ using negative residue representation ( $13 \equiv -4 \pmod{17}$ ):

13^3 \equiv (-4)^3 = -64 \pmod{17}

Divide $-64$ by 17:

-64 = 17 \times (-4) + 4

Thus:

-64 \equiv 4 \pmod{17}

Therefore, $13^{99} \pmod{17} = 4$ .

Final answer

Common trap

Reducing the exponent modulo 17 instead of modulo $\phi(17) = 16$ .

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