Number Theory | Integer Congruence Example 2

We give a few examples involving integer congruence and simplification, without using Fermat's little Theorem, Euler's theorem, or Wilson's theorem

[mohithraju]

Or we can use Wilson's theorem:
(p-1)! = -1 (mod p)
Where p is prime

[georgesadler]

Professor Penn, thank you for another classic example using Integer congruence in Number Theory.

[michaelempeigne]

why can't we use :


a^n = b^n ( mod p ) if a = b ( mod p )


so 28 = - 2 ( mod 10 )
that means that 28^13 = ( - 2 )^13 ( mod n )
28^13 = - 8192 ( mod 10 )
28^13 = - 2 mod ( 10 )
28^13 = 8 ( mod 10 )

[IODell]

Any suggestions on solving 3^45 (mod 45)? The solution is 18 but I'm stuck on method.

[mathisalwaysright]

Alternative sol: notice that (-2)^3 = -2 mod 10; so we only need to find (-2)^(13 mod 3) = -2 = 8 (mod 10)

[nilsastrup]

You wright equalsigns a lot even though the numbers are just congruent mod 10...

[adlynfakhreyza]

We also have wilson's theorem where :
(13-1)!= -1 (mod 13) = 12

[dzmitry-lahoda]

If you are not clear how things happen here, see Khan Academy Modular Arithmetic Course

[tomatrix]

Reduce exponent mod phi10 = phi5phi2 = 4 28 reduced mod 20 =8 so we get 8^1 =8

[وريانهاد]

(50-x)mod(1+7x)=0
How we deal with such kind of this equation ??? Good luck

[PunmasterSTP]

It seems like Huey Lewis was right; it's hip to be square!