In a basic encryption algorithm, a numerical value V is generated using the formula V = 47… — State PSC Exam

In a basic encryption algorithm, a numerical value $V$ is generated using the formula $V = 47^{91} - 19^{3}$ . The final security key is determined by the positive remainder when $V$ is divided by $12$ . What is the value of the final security key?

A
8
B
6
C
2
4Cevap

Cevap

The correct answer is found by applying modular arithmetic rules to reduce each part of the expression modulo 12. First,

47 \equiv -1 \pmod{12}

, so

47^{91} \equiv (-1)^{91} = -1

. Second,

19 \equiv 7 \pmod{12}

, and since

7^2 = 49 \equiv 1 \pmod{12}

, we get

7^3 \equiv 7 \times 1 = 7

. Subtracting these yields

-1 - 7 = -8

. To find the equivalent positive remainder, we add the divisor:

-8 + 12 = 4

Adım Adım Çözüm

Find the remainder of

47^{91}

modulo 12.

Since

47 = 12 \times 4 - 1

, it is easier to use

47 \equiv -1 \pmod{12}

. Therefore,

47^{91} \equiv (-1)^{91} = -1 \pmod{12}

Using negative remainders for numbers close to a multiple of the divisor simplifies large exponent calculations.

Find the remainder of

19^{3}

modulo 12.

Since

19 = 12 \times 1 + 7

, we have

19 \equiv 7 \pmod{12}

. Thus,

19^3 \equiv 7^3 \pmod{12}

. We know

7^2 = 49 \equiv 1 \pmod{12}

. Multiplying by 7 gives

7^3 \equiv 1 \times 7 = 7 \pmod{12}

Reducing the base before exponentiation makes the calculation manageable without needing to calculate

19^3

completely.

Subtract the individual remainders.

(-1) - 7 = -8 \pmod{12}

Applying modular arithmetic properties to combine the terms in the original expression.

Convert the negative remainder to a positive remainder.

-8 + 12 = 4

The standard remainder definition requires a non-negative integer strictly less than the divisor.

Anahtar Kavram

Modular arithmetic, negative remainders, and power reduction rules.

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