Examkin

Soru

Zorluk: OrtaDivisibility Rules and Remainder Theorem

What is the remainder when the value of the expression 240−5×32^{40} - 5 \times 3 is divided by 1010?

  1. A
    9
  2. 1Cevap
  3. C
    3
  4. D
    6

Cevap

1
The remainder when dividing by 10 is determined by the unit digit of the expression. First, evaluate the multiplication: 5×3=155 \times 3 = 15. Next, find the unit digit of 2402^{40}. The powers of 2 follow a cyclicity of 4 (2, 4, 8, 6). Since 40 is perfectly divisible by 4, the unit digit corresponds to the 4th value in the cycle, which is 6. Substituting these into the expression gives 6−15=−96 - 15 = -9. To find the valid positive remainder modulo 10, the divisor 10 must be added to the negative result: −9+10=1-9 + 10 = 1.

Adım Adım Çözüm

1
Evaluate the multiplication part of the expression.
The term 5×35 \times 3 equals 1515. The expression simplifies to 240−152^{40} - 15.
The order of operations (BODMAS) requires multiplication to be performed before subtraction.
2
Determine the unit digit (remainder modulo 10) of 2402^{40} by identifying the cyclicity of base 2.
The unit digits of powers of 2 repeat in a cycle of 4: 2, 4, 8, 6.
Since dividing by 10 is equivalent to finding the unit digit, we use power cyclicity to simplify large exponents.
3
Divide the exponent 40 by the cyclicity length 4.
40 divided by 4 leaves a remainder of 0. A remainder of 0 corresponds to the 4th position in the cycle.
The remainder determines the position in the cyclicity sequence. A remainder of 0 means the sequence has completed a full cycle.
4
Substitute the determined unit digit back into the simplified expression.
The 4th power in the cycle ends in 6. Substituting this into the expression yields 6−15=−96 - 15 = -9.
Replacing the large power with its unit digit equivalent allows us to calculate the raw remainder.
5
Convert the negative remainder into a valid positive remainder modulo 10.
−9+10=1-9 + 10 = 1. The final remainder is 1.
Remainders must be non-negative integers smaller than the divisor. Adding the divisor to a negative remainder provides the correct positive equivalent.

Anahtar Kavram

Applying cyclicity rules, the remainder theorem, and the order of operations to evaluate complex numerical expressions.
Bu soruyu puanla

Konular

Divisibility Rules and Remainder Theorem

Bu konudan daha fazla soru