Calculate the remainder when the integer value of 11 \times 17^{73} is divided by 19. — State PSC Exam

Calculate the remainder when the integer value of $11 \times 17^{73}$ is divided by $19$ .

Cevap: 16

Cevap

The correct remainder is found by applying Fermat's Little Theorem to reduce the large exponent, yielding an intermediate calculation of -22. Converting this negative value to a proper positive remainder modulo 19 gives exactly 16.

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Simplify the base of the exponent modulo 19.

17 \equiv -2 \pmod{19}

Working with a smaller absolute value simplifies subsequent exponentiation steps.

Apply Fermat's Little Theorem to identify the cyclicity.

a^{18} \equiv 1 \pmod{19}

for any integer

a

not divisible by 19.

Since 19 is a prime number, the remainders of powers will repeat in cycles of 18.

Reduce the large exponent

73

using the identified cyclicity.

73 = 18 \times 4 + 1

, which means

(-2)^{73} \equiv (-2)^1 \pmod{19}

Because the powers cycle every 18, only the remainder of the exponent divided by 18 affects the final result.

Multiply the reduced exponential term by the leading coefficient.

11 \times (-2) = -22

The original mathematical expression contains a coefficient of 11.

Convert the negative intermediate result to a valid positive remainder.

-22 \equiv -22 + 2 \times 19 = -22 + 38 = 16 \pmod{19}

By standard definition, a remainder must be a positive integer strictly less than the divisor.

Anahtar Kavram

Modular Arithmetic and Fermat's Little Theorem

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