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 >= 1 and m > 1. If a -= b (mod m) and c -= d (mod m), then a^c -= b^d (mod m)

sibuzwaW 2021-03-07 Answered
For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, d, mZ such that c, d1 and m > 1. If ab(modm) and
cd(modm), then acbd(modm)
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Expert Answer

ensojadasH
Answered 2021-03-08 Author has 100 answers
Let a, b, c, d, mZ such that c, d1 and m > 1.
If ab(modm)andcd(modm)
acmod(m)=(amod(m))c
bdmod(m)=(bmod(m))d
But ab(modm)
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 22(mod10)and166(mod10)
216¬26mod(10)
As 216=65536and26=32
65536-32 = 65504 which is not divisible by 10.
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