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Difficulty: MediumDivisibility Rules and Remainder Theorem

Consider the number P=4+7×3899P = 4 + 7 \times 38^{99}. What is the remainder when PP is divided by 1313?

  1. A
    3
  2. B
    11
  3. 10Answer
  4. D
    2

Answer

10
The correct remainder is found by applying modular arithmetic step-by-step. First, 3838 modulo 1313 simplifies to −1-1. Because 9999 is an odd power, (−1)99(-1)^{99} equals −1-1. Multiplying this by 77 yields −7-7. Adding 44 gives an intermediate remainder of −3-3. To find the true positive remainder, we add the divisor 1313 to −3-3, resulting in 1010.

Step-by-Step Solution

1
Simplify the base of the exponentiated term modulo 13.
38≡−1(mod13)38 \equiv -1 \pmod{13}
Using a negative remainder that is close to zero makes evaluating large exponential powers mathematically manageable.
2
Evaluate the exponentiated term modulo 13.
(−1)99=−1(-1)^{99} = -1
A negative base raised to an odd power retains its negative sign.
3
Substitute the simplified term back into the original expression and apply the correct order of operations.
4+7×(−1)=4−7=−34 + 7 \times (-1) = 4 - 7 = -3
Standard order of operations requires multiplication to be performed before addition.
4
Convert the negative intermediate remainder into a valid positive remainder.
−3≡−3+13=10(mod13)-3 \equiv -3 + 13 = 10 \pmod{13}
Remainders must be positive integers strictly less than the divisor.

Key Concept

Modular Arithmetic with Negative Bases and Order of Operations

Alternative Method

One could alternatively find the positive remainder of 38 modulo 13, which is 12, and then evaluate 4+7×1299(mod13)4 + 7 \times 12^{99} \pmod{13}. However, this requires observing the cyclicity pattern of powers of 12 modulo 13, making the negative base approach (-1) far more direct and efficient.
Estimated Time:1m 0s
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Topics

Divisibility Rules and Remainder Theorem

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