PROVE-THAT-6-8-x-2-x-

Question Number 170046 by Rasheed.Sindhi last updated on 14/May/22

PROVE THAT

6 ∣ (8^{x} - 2^{x})

Answered by JDamian last updated on 14/May/22

(8^{x} - 2^{x}) \mod 6 =

= 8^{x} \mod 6 - 2^{x} \mod 6 =

= {(8 \mod 6)}^{x} - 2^{x} \mod 6 =

= 2^{x} \mod 6 - 2^{x} \mod 6 =

= 0

Commented by Rasheed.Sindhi last updated on 15/May/22

T h a n X s i r!

Answered by mr W last updated on 15/May/22

8^{n} = {(2 + 6)}^{n} = \underset{k = 0}{\sum^{n}} C_{k}^{n} 6^{n - k} 2^{k} = \underset{k = 0}{\sum^{n - 1}} C_{k}^{n} 6^{n - k} 2^{k} + 2^{n}

8^{n} - 2^{n} = \underset{k = 0}{\sum^{n - 1}} C_{k}^{n} 6^{n - k} 2^{k}

\Rightarrow 6 ∣ 8^{n} - 2^{n}

Commented by Rasheed.Sindhi last updated on 15/May/22

T h a n X s i r!