"Decide whether each of these integers is congruent..." solved and explained

mentare9q

Open question

2022-08-18

Decide whether each of these integers is congruent to 3 modulo 7.
66

Saige Roy

Beginner2022-08-19Added 5 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 66 is larger than 3, we should be able to obtain 66 by consecutively adding 7 to 3 if

66 \equiv 3 b mod 7

3 b mod 7

\equiv 3 + 7 b mod 7

\equiv 10 b mod 7

\equiv 10 + 7 b mod 7

\equiv 17 b mod 7

\equiv 17 + 7 b mod 7

\equiv 24 b mod 7

\equiv 24 + 7 b mod 7

\equiv 31 b mod 7

\equiv 31 + 7 b mod 7

\equiv 38 b mod 7

\equiv 38 + 7 b mod 7

\equiv 45 b mod 7

\equiv 45 + 7 b mod 7

\equiv 52 b mod 7

\equiv 52 + 7 b mod 7

\equiv 59 b mod 7

\equiv 59 + 7 b mod 7

\equiv 66 b mod 7

We then obtained that 66 is congruent to

3 b mod 7

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