Quick modular arithmetic question (What is meant when two values are inverses of one another modular m)?

Question

So:

8x ≡ 5 (mod 13)

gcd (8,13) = 1
So use Euclidean algorithm

(-3) x 13 + 5 x 8 = 1

So 8 x 5 ≡ 1 (mod 13)

8 and 5 are inverses of one another modulo 13
What is meant by this?
Thank you :)

Answer 1

The inverse of a (mod m) is the number which you must multiply by a to be congruent to 1 (mod m). For instance, the inverse of 4 (mod 13) is 10 Because 4*10 = 40 = 1 (mod 13). Notice that this also necessarily implies that the inverse of 10 (mod 13) is 4. Inverses modulo m are unique just as they are over the rationals.



Also notice that a (mod m) exists iff (a,m) = 1. In general the easiest method for finding the inverse of a (mod m) (or for solving any linear congruence for that matter) is to continually add multiples of m to the right side of the equation until a multiple of a is found. Then divide by a to cancel on the left. For the above example we have that



10x = 1 (mod 13). We wish to find x, so we add 13 to the right until a multiple of 10 is found

10x = 14 (mod 13)

10x = 27 (mod 13)

10x = 40 (mod 13) <-----------multiple of 10. Now cancel 10 from both sides to get

x = 4 (mod 13).

Answer 2

8 and 5 are inverses under multiplication modulo 13 because 8 x 5 and 5 x 8 are equal to the identity element 1.

Answer 3

In standard arithmetic, the multiplicative inverse of a number a is the number b such that a*b=1, so b=1/a (the reciprocal). In modular arithmetic, there is no notion of "division", but the definition is still the same. The multiplicative inverse of a number a in mod m arithmetic is the number b such that a*b ≡ 1 mod m. (Normally, the definition of an inverse requires the identity to hold both ways ie. a*b ≡ 1 mod m and b*a ≡ 1 mod m, but since multiplication is commutative, you don't need to worry about that)



Hope it helps :)