An inventory tracking system logs the total number of specialized microchips stored in a w… — State PSC Exam

An inventory tracking system logs the total number of specialized microchips stored in a warehouse as exactly $5 \times 14^{33} - 3 \times 28^{50}$ units. These microchips are being shipped out in standard crates that hold exactly $13$ units each. The remaining microchips that do not make a full crate will be set aside for quality assurance testing. How many microchips will be set aside?

Answer: 6 microchips

Answer

6 microchips will be set aside for quality assurance testing.

By evaluating the modulo

13

for each term independently using exponent cyclicity, the expression

5 \times 14^{33}

reduces to

5

, and

3 \times 28^{50}

reduces to

12

. Subtracting these yields an intermediate remainder of

-7

. Since remainders representing physical quantities must be non-negative, we add the divisor (

13

) to

-7

, resulting in a final positive remainder of

6

Step-by-Step Solution

Formulate the modulo expression.

(5 \times 14^{33} - 3 \times 28^{50}) \pmod{13}

Finding the remaining chips after filling crates of 13 is equivalent to calculating the remainder when the total is divided by 13.

Reduce the base of the first term.

5 \times 1^{33} = 5 \pmod{13}

14

leaves a remainder of

1

when divided by

13

. Any power of

1

remains

1

Reduce the base of the second term.

3 \times 2^{50} \pmod{13}

28

divided by

13

2

with a remainder of

2

Apply Fermat's Little Theorem to evaluate

2^{50} \pmod{13}

2^{12} \equiv 1 \pmod{13}

, so

2^{50} \equiv 2^2 \equiv 4 \pmod{13}

Powers of

2

repeat every

12

modulo

13

. Since

50 = 12 \times 4 + 2

, the remainder of the exponent is

2

Evaluate the full second term.

3 \times 4 = 12 \pmod{13}

Multiply the evaluated exponent component by the original coefficient.

Subtract to find the initial remainder.

5 - 12 = -7

Substitute the evaluated remainders back into the full expression.

Convert the negative remainder into a positive value.

-7 + 13 = 6

A physical count must be a positive integer. Adding the divisor to a negative remainder gives the true positive remainder.

Key Concept

Modular arithmetic properties, Fermat's Little Theorem, and conversion of negative remainders.

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