Solved: "For the following statement, either prove that they..."

Emeli Hagan

Answered question

2020-11-12

For the following statement, either prove that they are true or provide a counterexample:
Let a, b, m,

n \in Z

such that m, n > 1 and

n ∣ m

. If

a \equiv b (mod m)

, then

a \equiv b (mod n)

aprovard

Skilled2020-11-13Added 94 answers

Given that,
Let a, b, m,

n \in Z

such that m, n > 1 and

n ∣ m

a \equiv b (mod m)

By using the definition of congruence relation,

a \equiv b (mod m)

means m divides b-a.

m ∣ (b - a)

Also given that

n ∣ m

,
By using,
If

a | b and b | c t h e n a ∣ c

.
Thus,

n | m and m | (b - a)

implies that

n ∣ (b - a)

.
By using reverse definition of congruence,

a \equiv b mod (n)

Therefore, if

a \equiv b (mod m) and n ∣ m

then

a \equiv b (mod n)

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