For the following statement, either prove that they are true or provide a counterexample: Let a, b, c, d, m∈Z such that c, d≥1 and m &gt; 1. If a≡b(modm) and c≡d(modm), then ac≡bd(modm)

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

High School
Geometry
High school geometry
Congruence

sibuzwaW

Answered question

2021-03-07

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

m \in Z

such that c,

d \geq 1

and m > 1. If

a \equiv b (mod m)

and

c \equiv d (mod m)

, then

a^{c} \equiv b^{d} (mod m)

Answer & Explanation

ensojadasH

Skilled2021-03-08Added 100 answers

Let a, b, c, d,

m \in Z

such that c,

d \geq 1

and m > 1.
If

a \equiv b (mod m) and c \equiv d (mod m)

a^{c} mod (m) = {(a mod (m))}^{c}

b^{d} mod (m) = {(b mod (m))}^{d}

But

a \equiv b (mod m)

So we get c = d.
Therefore the statement is true when c =d.
So take c and d are different for a counter example.
Take a = b = 2, c = 16 and d = 6
m=10
Hence

2 \equiv 2 (mod 10) and 16 \equiv 6 (mod 10)

2^{16} \neg \equiv 2^{6} mod (10)

2^{16} = 65536 and 2^{6} = 32

65536-32 = 65504 which is not divisible by 10.

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