What is the value of (17 - 42) \pmod{9} expressed in standard non-negative remainder form? — JAMB UTME

What is the value of $(17 - 42) \pmod{9}$ expressed in standard non-negative remainder form?

The value of

(17 - 42) \pmod{9}

in standard non-negative remainder form is

2

Evaluating

17 - 42

gives

-25

. Reducing

-25

modulo

9

requires finding a non-negative remainder

r

such that

-25 = 9q + r

with

0 \le r < 9

. Choosing

q = -3

yields

-25 = 9(-3) + 2

, making the correct non-negative remainder

2

. Alternatively,

17 \equiv 8 \pmod{9}

and

42 \equiv 6 \pmod{9}

, so

8 - 6 = 2 \pmod{9}

Perform the subtraction within the parentheses.

17 - 42 = -25

Evaluate the arithmetic expression inside the modulo operation first.

Express

-25

in terms of the modulus

9

using the division algorithm

a = qn + r

, where

0 \le r < n

-25 = 9 \times (-3) + 2

The remainder

r

must be non-negative (

0 \le r < 9

Identify the canonical non-negative remainder.

r = 2

Alternatively,

-25 \equiv -7 \pmod{9}

, and adding the modulus gives

-7 + 9 = 2

Modular Arithmetic and Negative Remainder Reduction

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