Suppose a certain positive integer x yields a remainder of 4 upon division by 17. What is … — State PSC Exam

Suppose a certain positive integer $x$ yields a remainder of $4$ upon division by $17$ . What is the remainder obtained when the quantity $5x - 38$ is divided by $17$ ?

A
$-1$
B
$0$
C
$1$
$16$ Answer

Answer

The correct answer is

16

By substituting the initial remainder

4

into the expression, we evaluate

5(4) - 38 = -18

. To find the valid positive remainder modulo

17

, we add multiples of

17

until the number becomes non-negative:

-18 + 34 = 16

Step-by-Step Solution

Identify the modular relationship for

x

x \equiv 4 \pmod{17}

The problem states that

x

leaves a remainder of

4

when divided by

17

Substitute this remainder into the given expression

5x - 38

5(4) - 38 = 20 - 38 = -18

Using the properties of modular arithmetic, we can substitute the remainder directly into polynomial expressions.

Find the equivalent positive remainder for

-18

modulo

17

-18 + 17 + 17 = 16

Remainders must be non-negative integers strictly less than the divisor. Adding multiples of

17

-18

gives the valid positive equivalent.

Key Concept

Modular Arithmetic and Negative Remainders

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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