Question

Difficulty: EasyModular Arithmetic

Find the value of $r$ such that $-29 \equiv r \pmod{8}$ , where $0 \le r < 8$ .

Answer: 3

The value of

r

in standard non-negative remainder form is

3

Dividing

-29

8

yields a quotient of

-4

and a remainder of

3

, because

8 \times (-4) + 3 = -29

. Since

0 \le 3 < 8

3

is the standard non-negative remainder.

Express

-29

in the form

8q + r

, where

q

is an integer and

0 \le r < 8

-29 = 8(-4) + 3

To find the standard remainder modulo

8

, the remainder

r

must satisfy

0 \le r < 8

Identify the remainder value

r

r = 3

Since

3

lies within the required range

[0, 7]

, it is the canonical remainder.

Modular Arithmetic and Non-Negative Remainders

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