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Difficulty: MediumDivisibility Rules and Remainder Theorem

Consider a sequence of integers generated by the mathematical formula Tn=17n−40T_n = 17^n - 40. What is the positive remainder obtained when the 123rd123^{\text{rd}} term of this sequence is divided by 99?

  1. 4Answer
  2. B
    5
  3. C
    6
  4. D
    3

Answer

4
Applying the rules of modular arithmetic, we first reduce the base 1717 modulo 99, giving −1-1. Raising −1-1 to the odd power of 123123 yields −1-1. The expression then simplifies to −1−40=−41-1 - 40 = -41. Because remainders must be non-negative, we add the next largest multiple of 99 (which is 4545) to −41-41, resulting in a final positive remainder of 44.

Step-by-Step Solution

1
Set up the expression for the 123rd123^{\text{rd}} term modulo 99.
T123≡(17123−40)(mod9)T_{123} \equiv (17^{123} - 40) \pmod 9
We need to find the remainder of the specific sequence term when divided by 99.
2
Reduce the base of the exponent modulo 99.
17≡−1(mod9)17 \equiv -1 \pmod 9
Using a base of −1-1 simplifies the calculation of large powers.
3
Evaluate the exponent part using the reduced base.
(−1)123=−1(-1)^{123} = -1
Raising a negative number to an odd power retains the negative sign.
4
Substitute back into the expression and compute the intermediate remainder.
−1−40=−41-1 - 40 = -41
Combining the evaluated power with the rest of the sequence formula.
5
Convert the negative value into a valid positive remainder modulo 99.
−41=9×(−5)+4  ⟹  Remainder is 4-41 = 9 \times (-5) + 4 \implies \text{Remainder is } 4
A true remainder must be a positive integer strictly less than the divisor. We achieve this by adding multiples of 99 until the value is positive.

Key Concept

Modular arithmetic reduction, exponent parity rules, and conversion of negative remainders.
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Topics

Divisibility Rules and Remainder Theorem

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