How do congruences work in mod arithmetic?

[mickles]

Hi, this is not a homework problem, i just have a hard time following the sequence of this

In the book , it shows a couple examples

(=== is the triple equal sign)

1) 26=19+7===3+7===10(mod8)
2) 15=19-4===3-4=-1(mod8)
3)38 = 19*2===3*2=6(mod8)
4) 7===2(mod5), 343=7^3===2^3=8(mod5)

I understand mod8 and whatnot, just how does the book go from 19+7 to 3+7, and from 19-4 to 3-4. I just don't get how 19 and 3 are logically connected

I see how 15===-1(mod8) and 26===10(mod8) and 38=6(mod8).

Any help understanding is appreciated

 

[Petek]

The examples you cited are using the following property of congruences: If a \equiv b \pmod{m}, then a + c \equiv b + c \pmod{m}. So, in your first example, let a = 19, b = 3, c = 7 and m = 8.

Does that help?

 

[ramsey2879]

Hi, this is not a homework problem, i just have a hard time following the sequence of this

In the book , it shows a couple examples

(=== is the triple equal sign)

1) 26=19+7===3+7===10(mod8)
2) 15=19-4===3-4=-1(mod8)
3)38 = 19*2===3*2=6(mod8)
4) 7===2(mod5), 343=7^3===2^3=8(mod5)

I understand mod8 and whatnot, just how does the book go from 19+7 to 3+7, and from 19-4 to 3-4. I just don't get how 19 and 3 are logically connected

I see how 15===-1(mod8) and 26===10(mod8) and 38=6(mod8).

Any help understanding is appreciated

Any number A === A - N Mod N Thus 19 === 11 === 3 Mod 8, Therefore 19+7 === 3+7 and 19-4 === 3-4.

 

[mickles]

The examples you cited are using the following property of congruences: If a \equiv b \pmod{m}, then a + c \equiv b + c \pmod{m}. So, in your first example, let a = 19, b = 3, c = 7 and m = 8.

Does that help?

Yes that makes a lot more sense now with a,b,c, and m after looking at the theorem.

Thanks for you help

 

[mickles]

Any number A === A - N Mod N Thus 19 === 11 === 3 Mod 8, Therefore 19+7 === 3+7 and 19-4 === 3-4.

thank you this also helped