A satellite transmission system encodes data packets using a cyclic redundancy check (CRC)… — State PSC Exam

A satellite transmission system encodes data packets using a cyclic redundancy check (CRC). The checksum value is determined by finding the remainder when the expression

T = 7 + 12 \times 53^{205}

is divided by

27

. What is the correct checksum value?

A
5
B
8
22Answer
D
19

Answer

The correct checksum value is 22.

The correct answer is 22. To find the remainder, we apply modular arithmetic to the base: since

27 \times 2 = 54

, we know

53 \equiv -1 \pmod{27}

. Substituting this into the expression gives

7 + 12(-1)^{205}

. Because the exponent 205 is an odd number,

(-1)^{205} = -1

. Applying the order of operations, we multiply 12 by -1 to get -12, and then add 7, resulting in

-5

. Finally, to convert a negative remainder into a positive one, we add the divisor:

-5 + 27 = 22

Step-by-Step Solution

Identify the base and express it in modular arithmetic relative to the divisor 27.

53 \equiv -1 \pmod{27}

Simplifying the base to a small integer makes evaluating large exponents manageable. Since

54

is a multiple of

27

53

is exactly

1

less than a multiple.

Evaluate the exponentiated term using the simplified base.

53^{205} \equiv (-1)^{205} \equiv -1 \pmod{27}

Raising a negative one to an odd exponent results in negative one.

Substitute this result back into the expression and apply the proper order of operations (BODMAS).

7 + 12 \times (-1) = 7 - 12 = -5 \pmod{27}

According to mathematical rules, multiplication must be performed before addition.

Convert the negative remainder into a valid positive remainder.

-5 + 27 = 22

A standard remainder must be a non-negative integer less than the divisor. We achieve this by adding the divisor to the negative result.

Key Concept

Modular Arithmetic and Fermat's/Euler's Remainder Concepts

Estimated Time:1m 15s

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