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## Homework Statement

Find x for

ax = 1 (mod m)

a) a = 15 , m = 31

b) a = 6, m = 93

c) a = 15, m = 20

if possible.

(The equal sign above is equivalence in modular arithmetic)

## Homework Equations

## The Attempt at a Solution

a) gcd(31,5) = gcd( 31 - 2*15, 15) = 1

thus (1)(31) + (-2)(15) = 1

and (-2) * 15 = 1 (mod 31)

i dont understand how I can find x since multiplying by -2 yields a number smaller than the modulus

b) gcd(93,6) = 3

no solution because gcd =/= 1

c) gcd(20,15) = 5

no solution

Could somebody please help me out with these problems. It's my first time doing modular arithmetic and the book has only two examples.

Thank you!

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