By DEFINITION to find the multiplication inverse of element `x` in `RR` means to find the integer number `y` which satisfies the relation
Therefore, to find the multiplication inverse of the integer element `x` in `ZZn` means to find the integer number `y < n` which satisfies the relation
`x*y =n*m +1` (1)
where `m` is also an integer (all numbers `x, y, n, m` are integers with `x` and `m` given)
We have `x =14` and `m=15` , and the above relation is
`14*y =15*m +1`
`14*y =(14+1)*m +1`
with the condition `y-m` is integer. Thus
`m+1 =14` or equivalent `m=13`
Now we can compute the value of `y`
`14*y =15*13 +1`
`y = 14`
Which means that the multiplication inverse of 14 in `ZZ15` is the number itself, 14.
In this case the relation (1) becomes
and the smallest integer numbers `y` and `m` that satisfy the above relation can be found by trial and error and observing that
`38*y ~~83*m` so `y/m ~~2` or equivalent `y~~2m` both numbers being integers.
Thus the above relation becomes true for `y=59` and `m=27` so that we can write
`38*59 =83*27 + 1`
Therefore the multiplication inverse of 38 in `ZZ83` is the number 59.