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Difficulty: MediumDivisibility Rules and Remainder Theorem

If the integer generated by the mathematical expression 15+35×368715 + 35 \times 36^{87} is divided by 3737, which of the following represents the correct positive remainder?

  1. A
    -20
  2. 17Answer
  3. C
    24
  4. D
    13

Answer

17
Applying modular arithmetic properties, we can evaluate each part of the expression relative to the divisor 3737. First, the base 3636 is equivalent to −1(mod37)-1 \pmod{37}. Raising −1-1 to the odd power of 8787 results in −1-1. Substituting this back into the equation yields 15+35×(−1)15 + 35 \times (-1). Following the standard order of operations, we multiply first to get −35-35, and then add 1515, resulting in an intermediate value of −20-20. Since remainders must be positive, adding the divisor (3737) to −20-20 produces the final valid remainder of 1717.

Step-by-Step Solution

1
Simplify the base 3636 modulo 3737.
36≡−1(mod37)36 \equiv -1 \pmod{37}
Reducing the base to a smaller equivalent modulus makes the large exponentiation manageable.
2
Evaluate the exponentiated term 3687(mod37)36^{87} \pmod{37}.
(−1)87=−1(-1)^{87} = -1
An odd power of a negative base retains the negative sign.
3
Substitute the reduced term back into the expression and apply the correct order of operations (BODMAS/PEMDAS).
15+35×(−1)=15−35=−2015 + 35 \times (-1) = 15 - 35 = -20
Multiplication must be resolved prior to addition.
4
Convert the intermediate negative remainder to a standard positive remainder.
−20+37=17-20 + 37 = 17
A final remainder must always be a positive integer strictly less than the divisor.

Key Concept

Modular Arithmetic with Negative Bases and Operational Order
Estimated Time:1m 15s
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Topics

Divisibility Rules and Remainder Theorem

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