In an advanced error-checking algorithm, a data packet's integrity is verified by calculat… — State PSC Exam

In an advanced error-checking algorithm, a data packet's integrity is verified by calculating a specific numerical signature. This signature is defined as the remainder when the value $V = 112^{2029} - 122^{2026}$ is divided by $13$ . What is the numerical signature of this data packet?

Cevap: 9

Cevap

The numerical signature of the data packet is 9.

The expression evaluates to

-4 \pmod{13}

after simplifying the bases and exponents. Converting this to a valid positive remainder by adding the modulus

13

yields the correct answer,

9

Adım Adım Çözüm

Reduce the base 112 modulo 13.

112 = 13 \times 8 + 8

, so

112 \equiv 8 \pmod{13}

In modular arithmetic, bases can be replaced by their remainders to simplify exponentiation.

Reduce the base 122 modulo 13.

122 = 13 \times 9 + 5

, so

122 \equiv 5 \pmod{13}

This simplifies the second term of the expression.

Apply Fermat's Little Theorem to simplify the exponent for the first term.

Since 13 is prime,

a^{12} \equiv 1 \pmod{13}

. For

8^{2029}

, dividing

2029

12

leaves a remainder of

1

(

2029 = 12 \times 169 + 1

). Thus,

8^{2029} \equiv 8^1 = 8 \pmod{13}

Large powers can be reduced by finding the remainder of the exponent divided by

p-1

Apply Fermat's Little Theorem to simplify the exponent for the second term.

For

5^{2026}

, dividing

2026

12

leaves a remainder of

10

(

2026 = 12 \times 168 + 10

). Thus,

5^{2026} \equiv 5^{10} \pmod{13}

This reduces the exponent to a manageable size.

Evaluate

5^{10} \pmod{13}

Since

5^2 = 25 \equiv -1 \pmod{13}

, we calculate

5^{10} = (5^2)^5 \equiv (-1)^5 = -1 \equiv 12 \pmod{13}

Using negative remainders for intermediate steps significantly speeds up manual exponentiation.

Subtract the evaluated terms and find the final positive remainder.

8 - 12 = -4

. To find the positive remainder, add the modulus 13:

-4 + 13 = 9

The final remainder must be a non-negative integer strictly less than the divisor.

Anahtar Kavram

Modular arithmetic, Fermat's Little Theorem, and handling negative remainders.

Alternatif Yöntem

Instead of using Fermat's Little Theorem (mod 12), you can find the exact order of the bases. Notice that

8^2 = 64 \equiv -1 \pmod{13}

, so

8^4 \equiv 1 \pmod{13}

. Similarly,

5^2 = 25 \equiv -1 \pmod{13}

, so

5^4 \equiv 1 \pmod{13}

. Both bases have an order of 4 modulo 13. Dividing the exponents by 4:

2029 \div 4

leaves a remainder of 1, so

8^{2029} \equiv 8^1 = 8

2026 \div 4

leaves a remainder of 2, so

5^{2026} \equiv 5^2 = 25 \equiv -1 \equiv 12

. The final subtraction is

8 - 12 = -4 \equiv 9 \pmod{13}

. This method requires less calculation.

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