What is the positive remainder when the product 54 \times 82 is divided by 11? — State PSC Exam

What is the positive remainder when the product $54 \times 82$ is divided by $11$ ?

Answer: 6

By finding the individual remainders of the factors (

-1

and

5

) and multiplying them, we get

-5

. Adding the divisor (

11

) to this negative remainder yields the correct positive remainder of

6

Find the remainder of

54

when divided by

11

Since

11 \times 5 = 55

, we can write

54

55 - 1

. Thus,

54 \equiv -1 \pmod{11}

Using a negative remainder simplifies the subsequent multiplication.

Find the remainder of

82

when divided by

11

Since

11 \times 7 = 77

, we have

82 - 77 = 5

. Thus,

82 \equiv 5 \pmod{11}

We need the remainder of the second factor to apply the remainder theorem for products.

Multiply the individual remainders.

(-1) \times 5 = -5

According to modular arithmetic properties, the remainder of a product is the product of the individual remainders.

Convert the negative remainder to a positive remainder.

-5 + 11 = 6

The question specifically asks for the positive remainder. Adding the divisor to a negative remainder gives the equivalent positive remainder.

Remainder Theorem and Modular Arithmetic

Question