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Difficulty: EasyDivisibility Rules and Remainder Theorem

An arithmetic expression is given as 20×26−4020 \times 26 - 40. Which of the following represents the positive remainder when the final computed value of this expression is divided by 77?

  1. A
    0
  2. B
    3
  3. 4Answer
  4. D
    6

Answer

The correct positive remainder is 4.
By following the order of operations, the expression evaluates to 480. When 480 is divided by 7, the quotient is 68, leaving a positive remainder of 4 (68×7=47668 \times 7 = 476, and 480−476=4480 - 476 = 4). The same result is efficiently reached using modular arithmetic: (6×5)−5=25(6 \times 5) - 5 = 25, and 25≡4(mod7)25 \equiv 4 \pmod 7.

Step-by-Step Solution

1
Calculate the exact numerical value of the expression.
20×26−40=520−40=48020 \times 26 - 40 = 520 - 40 = 480.
Applying standard order of operations (BODMAS), multiplication is performed before subtraction.
2
Divide the computed value by 7 to find the remainder.
480÷7=68480 \div 7 = 68 with a remainder of 44.
The remainder is the integer left over after dividing the total by the divisor.
3
Verify using modular arithmetic (optional).
20≡−1(mod7)20 \equiv -1 \pmod 7, 26≡5(mod7)26 \equiv 5 \pmod 7, and 40≡5(mod7)40 \equiv 5 \pmod 7. Thus, (−1×5)−5=−10(-1 \times 5) - 5 = -10. Modulo 7, −10≡−3≡4-10 \equiv -3 \equiv 4.
Modular arithmetic provides a faster secondary method to confirm the remainder without computing large numbers.

Key Concept

Divisibility Rules and Remainder Theorem
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Topics

Divisibility Rules and Remainder Theorem

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