In a centralized database architecture, data records are distributed across 17 server node… — State PSC Exam

In a centralized database architecture, data records are distributed across $17$ server nodes numbered from $0$ to $16$ . A record with a numerical key $K$ is assigned to a node using the hash function $H(K) = K \pmod{17}$ , which always yields a positive remainder. If a specific batch of records is assigned a master key given by the expression $K = 2^{200} - 75$ , which server node will process this batch?

Cevap: 11

Cevap

The batch of records will be processed by server node 11.

By applying modular arithmetic rules, the expression

(2^{200} - 75) \pmod{17}

simplifies individually to

(1 - 7) = -6

. Because node numbers in this system must be positive, adding the divisor 17 to -6 provides the correct positive remainder, which is 11.

Adım Adım Çözüm

Identify the mathematical goal of the problem.

Calculate

(2^{200} - 75) \pmod{17}

and ensure the final result is a positive integer between 0 and 16.

The server nodes are assigned strictly based on the positive remainder when the key is divided by 17.

Find the remainder of the exponential term

2^{200}

divided by 17.

2^4 = 16 \equiv -1 \pmod{17}

. Raising this to the 50th power gives

(2^4)^{50} \equiv (-1)^{50} = 1 \pmod{17}

Finding a power of the base that is 1 or -1 modulo the divisor simplifies extremely large exponent calculations.

Find the remainder of the constant term 75 divided by 17.

75 = 17 \times 4 + 7

, so

75 \equiv 7 \pmod{17}

Each term in a modular arithmetic expression can and should be reduced individually.

Combine the reduced terms to find the overall remainder.

1 - 7 = -6 \pmod{17}

Substitute the individual reduced remainders back into the original expression

K = 2^{200} - 75

Convert the negative remainder into a valid positive node number.

-6 + 17 = 11

A remainder of -6 indicates the value is 6 units short of a full multiple of 17. The equivalent positive remainder is found by adding the divisor.

Anahtar Kavram

Modular arithmetic with large exponents and handling negative remainders appropriately.

Tahmini Süre:1m 30s

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