When a positive integer x is divided by 17, the remainder is 12. When another positive int… — State PSC Exam

When a positive integer $x$ is divided by $17$ , the remainder is $12$ . When another positive integer $y$ is divided by $17$ , the remainder is $15$ . What is the positive remainder when the expression $4x - 5y$ is divided by $17$ ?

A
4
7Answer
C
10
D
16

Answer

Using the properties of modular arithmetic, we substitute the remainders into the expression:

4(12) - 5(15) = 48 - 75 = -27

. To find the equivalent positive remainder modulo

17

, we add the nearest larger multiple of

17

, which is

34

. Thus,

-27 + 34 = 7

Step-by-Step Solution

Express the given properties using modular arithmetic.

x \equiv 12 \pmod{17}

and

y \equiv 15 \pmod{17}

This allows us to substitute the remainders directly into the expression.

Substitute the remainders into the expression

4x - 5y

4(12) - 5(15)

By the properties of modular arithmetic, the remainder of a sum or difference is the sum or difference of the individual remainders.

Evaluate the arithmetic expression.

48 - 75 = -27

Simplifying the multiplication and subtraction gives the intermediate negative remainder.

Convert the negative remainder into a valid positive remainder.

-27 \equiv -27 + 2(17) \equiv -27 + 34 \equiv 7 \pmod{17}

A valid remainder must be a positive integer between

0

and the divisor minus

1

Key Concept

Modular Arithmetic Operations and Negative Remainders

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