An industrial control system tracks part serial numbers using a cyclic validation algorith… — State PSC Exam

An industrial control system tracks part serial numbers using a cyclic validation algorithm. A batch identifier $N$ is generated using the formula $N = 65^{202} - 32^{145} - 45$ . The system validates this batch by computing the positive remainder when $N$ is divided by the base modulus $33$ . What is the expected positive remainder for this batch?

Answer: 23

Answer

The expected positive remainder is 23.

By applying the principles of modular arithmetic, we can simplify each term of the expression

65^{202} - 32^{145} - 45

modulo

33

. First,

65 \equiv -1 \pmod{33}

, so

65^{202} \equiv (-1)^{202} = 1

. Next,

32 \equiv -1 \pmod{33}

, so

32^{145} \equiv (-1)^{145} = -1

. Substituting these simplified values back into the expression yields

1 - (-1) - 45 = 2 - 45 = -43

. To find the positive remainder when dividing by

33

, we add a multiple of

33

(in this case,

66

) to

-43

, which results in

23

Step-by-Step Solution

Express the base numbers

65

and

32

modulo

33

65 \equiv -1 \pmod{33}

and

32 \equiv -1 \pmod{33}

Finding remainders close to

1

-1

significantly simplifies the evaluation of large exponents.

Evaluate the first term,

65^{202}

modulo

33

(-1)^{202} = 1

An even power of a negative one results in positive one.

Evaluate the second term,

32^{145}

modulo

33

(-1)^{145} = -1

An odd power of a negative one results in negative one.

Substitute the simplified terms back into the original expression modulo

33

N \equiv 1 - (-1) - 45 \equiv 2 - 45 \equiv -43 \pmod{33}

Combines all parts of the expression to find the overall remainder before final normalization.

Convert the negative remainder into a valid positive remainder.

-43 \equiv -43 + 66 = 23 \pmod{33}

Remainders must be positive integers between

0

and the modulus minus one. Adding the smallest multiple of

33

that is greater than

43

(which is

66

) gives the valid positive remainder.

Key Concept

Modular Arithmetic and Remainder Theorem with Negative Remainders

Alternative Method

You could also simplify the constant term first:

-45 \equiv -12 \equiv 21 \pmod{33}

. Then the expression becomes

1 - (-1) + 21 = 23

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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