Find the multiplication inverses of the given elements. (a) [14] in Z15 (b) [38] in Z83

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valentin68 | College Teacher | (Level 3) Associate Educator

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By DEFINITION to find the multiplication inverse of element `x` in `RR` means to find the integer number `y` which satisfies the relation

`x*y =1`

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)

A)

We have `x =14` and `m=15` , and the above relation is

`14*y =15*m +1`

`14*y =(14+1)*m +1`

`14*(y-m) =m+1`

`y-m =(m+1)/14`

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.

B)

In this case the relation (1) becomes

`38*y =83*m+1`

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.

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