Examkin

Soru

Zorluk: KolayDivisibility Rules and Remainder Theorem

What is the remainder when 62×5362 \times 53 is divided by 6060?

  1. A
    14
  2. 46Cevap
  3. C
    55
  4. D
    9

Cevap

46
To find the remainder of a product, we can multiply the remainders of each individual number. When divided by 60, 62 leaves a remainder of 2. So, we multiply 2 by 53 to get 106. Finally, we divide 106 by 60, which leaves a remainder of 46.

Adım Adım Çözüm

1
Find the remainder of each factor when divided by 60.
62≡2(mod60)62 \equiv 2 \pmod{60} and 53≡53(mod60)53 \equiv 53 \pmod{60} (or −7(mod60)-7 \pmod{60}).
By the properties of modular arithmetic, we can simplify large multiplications by replacing numbers with their remainders.
2
Multiply the remainders together.
2×53=1062 \times 53 = 106. Alternatively, using negative remainders: 2×(−7)=−142 \times (-7) = -14.
The remainder of a product is equal to the product of the individual remainders.
3
Find the final remainder by dividing the product from Step 2 by 60.
106=60×1+46106 = 60 \times 1 + 46, so the remainder is 4646. (If using −14-14, add 6060 to get 4646).
The intermediate product must be reduced modulo 60 to find the final positive remainder.

Anahtar Kavram

Properties of Remainders in Multiplication
Bu soruyu puanla

Konular

Divisibility Rules and Remainder Theorem

Bu konudan daha fazla soru