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Question

Difficulty: EasyDivisibility Rules and Remainder Theorem

In a sequence of arithmetic operations, a candidate calculates a value of −82-82. If this integer is then divided by 99, determine the least non-negative remainder.

Answer: 8

Answer

8
Applying the division algorithm, −82-82 can be expressed in the form dq+rdq + r as 9×(−10)+89 \times (-10) + 8. Since the remainder rr must be greater than or equal to 00 and strictly less than the divisor 99, the least non-negative remainder is 88.

Step-by-Step Solution

1
Identify the dividend and the divisor.
Dividend is −82-82, divisor is 99.
To apply the formal mathematical definition of division with remainders.
2
Find the largest integer multiple of 99 that is less than or equal to −82-82.
The correct multiple is −90-90 (since −90≤−82-90 \leq -82).
The remainder must be a strictly non-negative value, meaning the multiple we subtract from the dividend must not exceed the dividend.
3
Calculate the difference between the dividend and this multiple to find the remainder.
−82−(−90)=8-82 - (-90) = 8.
This difference represents the least non-negative remainder required by the problem.

Key Concept

Remainder Theorem and Negative Dividends
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Topics

Divisibility Rules and Remainder Theorem

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