I have been calculating Modular arithmetic for 2 by 10 but I don't understand what happened

Asked Jan 19 '22 at 00:48

Active Jan 19 '22 at 08:01

Viewed 52 times

So I have been calculating $2^{n}$ by $10$ until I find $n$ where n is a natural number.

Basically, I have trying values of $n$ from $0$ onward until I find the number where it keeps repeating.

But here is the thing:

$2^{0} \equiv 1 \pmod {10}$

$2^{1} \equiv 2 \pmod {10}$

$2^{2} \equiv 4 \pmod {10}$

$2^{3} \equiv 8 \pmod {10}$

$2^{4} \equiv 6 \pmod {10}$

But then this happens and I don't get it

$2^{5} \equiv 2 \pmod {10}$

Remainder is $2$ even though it should have been $1$ since it's like a clock, it should keep repeating itself from the very beginning but it started from the the second one not the first.

I don't get it. Can anybody explain this to me?

edited Jan 19 '22 at 08:01

Bill Dubuque

272,048

asked Jan 19 '22 at 00:48

TechnoKnight

2

No power of 2 is odd. – Sean Roberson Jan 19 '22 at 00:49
@SeanRoberson It's actually 32 but by using modular arithmetic properties, you can make it 2(32 - 3(10) – TechnoKnight Jan 19 '22 at 00:51
2

$\gcd(2,10)>1$, so no power of $2$ will ever be $1\pmod {10}$. – lulu Jan 19 '22 at 00:52
@lulu but $2^{0}$ is 1 (mod 10) – TechnoKnight Jan 19 '22 at 00:53
1

Yes, I meant $2^n$ for $n>0$. – lulu Jan 19 '22 at 00:54
@lulu okay but I thought modular arithmetic are supposed to be like a clock. Like they keep repeating themselves. I'm just so confused because I never, ever encountered something like this before and I solved many many of these modular arithmetic problems before. – TechnoKnight Jan 19 '22 at 00:56
2

The powers of $2$ do repeat themselves $\pmod {10}$, just after an initial non-repeating block. Perhaps you are more accustomed to working with powers of a number which is coprime to the modulus. – lulu Jan 19 '22 at 00:58
@lulu That must be it then. Thanks btw! – TechnoKnight Jan 19 '22 at 00:59
Or maybe you’ve been doing additive problems. Multiplication has quirks that don’t show up in additive behavior, as you’re seeing here. – Lubin Jan 19 '22 at 05:39

I have been calculating Modular arithmetic for 2 by 10 but I don't understand what happened

0 Answers0