Multiplicative inverse method of calculating

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Inverse definition

According to the previous definition: For the modulus \ (m \) , if for the remaining class \ ([A] \) , the presence of the remaining class \ ([B] \) such that \ ([a] \ cdot [ b] = [1 ] \) is called \ ([B] \) of \ ([a] \) of the inverse element

We written in the form congruence: if for modulus \ (m \) and the integer \ (A \) , if the \ (b \ cdot a \ equiv 1 (\ mod m) \) is called \ (B \) is \ (a \) of the inverse element, referred to as \ (a ^ {- 1} \)

The PEI genus theorem, only \ (ax + my = k \ cdot gcd (a, m), k \ in Z _ + \) solvable

Therefore, only \ (ax \ equiv k \ cdot gcd (a, m) (\ mod m) \) is solvability

Thus, \ (A \) Sufficient Conditions for inverse \ (gcd (a, m) = 1 \)