Solving Modulus Math Problems with Negative Numbers | Step-by-Step Guide

[Fairy111]

Homework Statement

Im confused as to how you obtain that,

77 is congruent to -1 mod 26
-77 is congruent to 1 mod 26
-11 is congruent to 15 mod 26

Homework Equations

The Attempt at a Solution

Some help would be great thanks

 

[lanedance]

haven't worked a heap with mod, but i think the trick is that the remainder is positive, so for the first
77 = 2x26 + 25, so 77mod26 = 25

simlarly
-1 = (-1)*26 + 25, then (-1)mod26 = 25

 

[Dick]

a is congruent to b mod 26 if a-b is divisible by 26. 77-(-1)=78, -77-1=(-78), -11-15=(-26). All of the differences are evenly divisible by 26.

 

[lanedance]

to clrify, i think mod is defined by the fact that if
a mob n = b
then (b-a) is a multiple of n

so take a poitive number c, such that c<n, clearly
c mod n = c
now consider the negative case, say
(-c) mod n = b
then as c<n, and b>0, we must have
(-c)-b = (-1)n
so
b = n -c