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

dejanimaab

Open question

2022-08-20

Decide whether each of these integers is congruent to 3 modulo 7.
37

Annalise Baldwin

Beginner2022-08-21Added 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 37 is larger than 3, we should be able to obtain 37 by consecutively 7 to 3 if

37 \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

We then note that 3\bmod 7 is equivalent with 31 and 38.

3 b mod 7

is then not equivalent with 37, since 31<37<38.
Note

37 b mod 7 \equiv 2 b mod 7

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