easy proof
**using congruences** show that $\large 2^{4x} \equiv 6 \mod 10 $

Question

easy proof
**using congruences** show that $\large  2^{4x} \equiv 6 \mod 10 $

dan815 · Answer

ohh

ganeshie8 · Answer

for example :
$\large 2^{4*349850934850943850439}$ leaves a remainder $\large 6$ when divided by $\large 10$

dan815 · Answer

2^4x probably has some pattern not the same as showing it must be with mod stuff tho lol

dan815 · Answer

like ending in 16 or

dan815 · Answer

26 36 46...

ganeshie8 · Answer

exactly! 16*16*16.... always leaves the remainder 6
the question was to prove it using congruences

dan815 · Answer

2
4
8
16<--- 6
32
64
128
256<------
512
1024
2048
4096<----------

How to show with modularity that every 4th power of 2 must end in 6

anonymous · Answer

2^(4x) = (2^4)^x = (last digit is 6)^x

dan815 · Answer

ok lol

ganeshie8 · Answer

a = b mod 10

doesn't mean a^k = b mod 10, right ?

anonymous · Answer

and last digit of  6^n is always 6

dan815 · Answer

that was too simple haha xD

ganeshie8 · Answer

Oh okay i see...

ikram002p · Answer

but why 6^n mod 10 always 6 :P

dan815 · Answer

imagine the long multiplication

ikram002p · Answer

i mean using congrunence *

ganeshie8 · Answer

you need to ask that question in abstract algebra/analysis course, 
not in number theory :P

dan815 · Answer

what does it mean to use congruence?

ikram002p · Answer

no its in number theory :o

abstract has nothing to do with it

ganeshie8 · Answer

using congruences means to use modular arithmetic properties

ikram002p · Answer

prove that

6^n= 6 mod 10

6=5+1 maybe work ?

ganeshie8 · Answer

binomial expansion ?

ikram002p · Answer

yep , then mod it

ikram002p · Answer

lol what a word "mod it " xD

anonymous · Answer

Since 6^2 MOD 10 =6 , 
6^n MOD 10 = 6

ikram002p · Answer

i hope the gost understand this as well :3


@sauravshakya  mmm not convising

ganeshie8 · Answer

A^2 = A

=>

A^n = A

ikram002p · Answer

since when :0

ganeshie8 · Answer

since i took LA :)
think of idempotent matrices

ikram002p · Answer

3^2 =9 mod 10

3^3=7 mod 10

counterexample

ikram002p · Answer

xD teach me LA

ganeshie8 · Answer

if 3^2 = 3 mod  something,
then definitely 3^n  = 3 mod samething.

ikram002p · Answer

ohh

ok got what u mean

ikram002p · Answer

wondering what dan was typing :3

ikram002p · Answer

nice one :o

dan815 · Answer

how come we cant just say that the starting playing must always be a 6

dan815 · Answer

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