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Difficulty: EasyDivisibility Rules and Remainder Theorem

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

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

Answer

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.

Step-by-Step Solution

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.

Key Concept

Properties of Remainders in Multiplication
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Topics

Divisibility Rules and Remainder Theorem

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