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Difficulty: MediumDivisibility Rules and Remainder Theorem

A manufacturing plant produces a total of 10+35×414310 + 35 \times 41^{43} small electronic components. These components are to be distributed equally into shipping containers that hold exactly 4242 units each. How many components will be left unpacked after filling as many containers as possible?

  1. A
    25
  2. B
    39
  3. 17Answer
  4. D
    3

Answer

17
To find the leftover components, we must calculate the remainder of the total amount modulo 42. Using modular arithmetic, 41 is congruent to -1 modulo 42. Raising -1 to the odd power of 43 yields -1. Multiplying this by 35 gives -35. Adding 10 results in -25. Because a physical remainder must be positive, we add the divisor (42) to the intermediate value of -25 to find the equivalent positive remainder, which is 17.

Step-by-Step Solution

1
Set up the modulo equation to find the remainder of the total components divided by 42.
(10 + 35 \times 41^{43}) \pmod{42}
The number of unpacked components represents the mathematical remainder when the total quantity is divided by the container capacity.
2
Simplify the large base (41) with respect to the divisor (42).
41 \equiv -1 \pmod{42}
Replacing a large base with its smallest absolute congruent value simplifies large exponent calculations.
3
Substitute -1 into the expression and evaluate the odd exponent.
10 + 35 \times (-1)^{43} \equiv 10 + 35 \times (-1) \pmod{42}
A negative base raised to an odd power evaluates to a negative result.
4
Perform the multiplication and addition following standard order of operations.
10 - 35 = -25 \pmod{42}
Multiplication must be completed strictly before addition.
5
Convert the negative remainder into a valid strictly positive remainder.
-25 + 42 = 17
Physical remainders cannot be negative. To find the equivalent positive remainder in modular arithmetic, add the divisor to the negative result.

Key Concept

Modular Arithmetic and Negative Remainders
Estimated Time:1m 30s
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Topics

Divisibility Rules and Remainder Theorem

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