What is the value of (-38) \pmod{7} expressed in standard non-negative remainder form? — JAMB UTME

What is the value of $(-38) \pmod{7}$ expressed in standard non-negative remainder form?

The value

4

is correct because

-38 = 7 \times (-6) + 4

. The remainder

4

lies in the standard non-negative range

0 \le r < 7

. Alternatively, adding multiples of

7

-38

gives

-38 + 35 = -3

, and adding

7

once more gives

-3 + 7 = 4

Find the largest multiple of the modulus

7

that is less than or equal to

-38

The multiple is

7 \times (-6) = -42

Modular arithmetic requires the remainder

r

to satisfy

0 \le r < 7

in standard form.

Calculate the remainder by subtracting the multiple from the dividend.

-38 - (-42) = -38 + 42 = 4

The remainder is the non-negative difference between the number and the multiple of the modulus.

Modular Arithmetic with Negative Numbers

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