Best solutions to - "Decide whether each of these integers is congruent..."

cvnoticiasdg

Open question

2022-08-19

Decide whether each of these integers is congruent to 3 modulo 7.
-67

Paola Mercer

Beginner2022-08-20Added 11 answers

Definitions
Division algorithm Let a be an integer and d a positive integer. Then there are nique integers q and r with

0 \leq r < d

such that a=dq+r
q is called the quotient and r is called the remainder

q = a \div d

r = a b mod d

Solution

3 b mod 7

Since -67 is smaller than 3, we should be able to obtain -67 by consecutively subtracting 7 from 3 if

- 67 \equiv 3 b mod 7

3 b mod 7

\equiv 3 - 7 b mod 7

\equiv - 4 b mod 7

\equiv - 4 - 7 b mod 7

\equiv - 11 b mod 7

\equiv - 11 - 7 b mod 7

\equiv - 18 b mod 7

\equiv - 18 - 7 b mod 7

\equiv - 25 b mod 7

\equiv - 25 - 7 b mod 7

\equiv - 32 b mod 7

\equiv - 32 - 7 b mod 7

\equiv - 39 b mod 7

\equiv - 39 - 7 b mod 7

\equiv - 46 b mod 7

\equiv - 46 - 7 b mod 7

\equiv - 53 b mod 7

\equiv - 53 - 7 b mod 7

\equiv - 60 b mod 7

\equiv - 60 - 7 b mod 7

\equiv - 67 b mod 7

We then obtained that -67 is congruent to

3 b mod 7

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