An integer P is given by the mathematical expression P = 4 \cdot 36^{2033} - 6 \cdot 58^{2… — State PSC Exam

An integer $P$ is given by the mathematical expression $P = 4 \cdot 36^{2033} - 6 \cdot 58^{2033} - 2 \cdot 89^{2033}$ . Determine the positive remainder when $P$ is divided by $17$ .

Answer: 9

Answer

The correct answer is derived by first reducing the bases modulo 17, giving 2, 7, and 4 respectively. Then, utilizing Fermat's Little Theorem (

a^{16} \equiv 1 \pmod{17}

), the exponent 2033 is reduced to

2033 \pmod{16} = 1

. Substituting these simplified values back into the expression yields

4(2) - 6(7) - 2(4) = -42

. Because a remainder must be positive, adding the next highest multiple of 17 (which is

17 \times 3 = 51

) to -42 gives the final valid remainder of 9.

Step-by-Step Solution

Find the remainder of each base when divided by 17.

36 \equiv 2 \pmod{17}

58 \equiv 7 \pmod{17}

, and

89 \equiv 4 \pmod{17}

Reducing bases before dealing with large exponents simplifies the modular arithmetic calculation.

Apply Fermat's Little Theorem to simplify the exponent.

Since 17 is a prime number,

a^{16} \equiv 1 \pmod{17}

. The exponent is

2033 = 16 \times 127 + 1

, meaning

a^{2033} \equiv a^1 \pmod{17}

Fermat's Little Theorem allows us to reduce massive exponents by dividing them by

(p-1)

and keeping only the remainder.

Substitute the reduced bases and exponents into the original expression.

P \equiv 4(2^1) - 6(7^1) - 2(4^1) \pmod{17}

, which evaluates to

8 - 42 - 8 = -42

This step calculates the combined overall remainder before adjusting for the strict definition of a positive modulo.

Convert the negative result to a valid positive remainder.

-42 \equiv -42 + 51 = 9 \pmod{17}

A remainder must be a non-negative integer strictly less than the divisor (17). Adding the next highest multiple of 17 (which is 51) provides the correct positive remainder.

Key Concept

Divisibility Rules and Remainder Theorem

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