A computer algorithm processes a data block and assigns it a routing ID based on modular a… — State PSC Exam

A computer algorithm processes a data block and assigns it a routing ID based on modular arithmetic. This ID corresponds to the strictly positive remainder left after dividing the value of $56^{103} - 45$ by $13$ .

Find the value of this routing ID.

Cevap: 11

Cevap

By reducing each component of the expression modulo 13,

56^{103}

reduces to

4^{103}

, which simplifies to

4

because

4^3 \equiv -1 \pmod{13}

. Subtracting

45 \pmod{13}

, which is

6

, yields

-2

. Adding the divisor 13 to

-2

gives the strictly positive remainder of 11.

Adım Adım Çözüm

Reduce the base 56 modulo 13.

56 \equiv 4 \pmod{13}

Modular arithmetic allows simplifying the base before exponentiation.

Find a pattern to easily evaluate

4^{103} \pmod{13}

4^3 = 64 \equiv -1 \pmod{13}

Finding a small power that is congruent to 1 or -1 simplifies large exponents significantly.

Evaluate

4^{103} \pmod{13}

using the established pattern.

4^{103} = (4^3)^{34} \times 4^1 \equiv (-1)^{34} \times 4 = 4 \pmod{13}

Exponent rules allow breaking down 103 into

3 \times 34 + 1

Reduce the subtracted constant 45 modulo 13.

45 = 13 \times 3 + 6 \equiv 6 \pmod{13}

All terms in the expression must be evaluated under the same modulus.

Subtract the reduced values.

4 - 6 = -2

Substitute the individual modulo results back into the original expression structure.

Convert the negative remainder into a positive remainder.

-2 + 13 = 11

The question asks for a strictly positive remainder, which is standard in modular division contexts.

Anahtar Kavram

Modular Arithmetic and Negative Remainders

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