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Zorluk: KolayDivisibility Rules and Remainder Theorem

Suppose a certain positive integer xx yields a remainder of 44 upon division by 1717. What is the remainder obtained when the quantity 5x−385x - 38 is divided by 1717?

  1. A
    −1-1
  2. B
    00
  3. C
    11
  4. 1616Cevap

Cevap

The correct answer is 1616.
By substituting the initial remainder 44 into the expression, we evaluate 5(4)−38=−185(4) - 38 = -18. To find the valid positive remainder modulo 1717, we add multiples of 1717 until the number becomes non-negative: −18+34=16-18 + 34 = 16.

Adım Adım Çözüm

1
Identify the modular relationship for xx
x≡4(mod17)x \equiv 4 \pmod{17}
The problem states that xx leaves a remainder of 44 when divided by 1717.
2
Substitute this remainder into the given expression 5x−385x - 38
5(4)−38=20−38=−185(4) - 38 = 20 - 38 = -18
Using the properties of modular arithmetic, we can substitute the remainder directly into polynomial expressions.
3
Find the equivalent positive remainder for −18-18 modulo 1717
−18+17+17=16-18 + 17 + 17 = 16
Remainders must be non-negative integers strictly less than the divisor. Adding multiples of 1717 to −18-18 gives the valid positive equivalent.

Anahtar Kavram

Modular Arithmetic and Negative Remainders
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Konular

Divisibility Rules and Remainder Theorem

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