In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. (wikipedia.org)
Furthermore, any integer that is congruent to a (i.e., in a's congruence class) has any element of x's congruence class as a modular multiplicative inverse. (wikipedia.org)
If a ≡ 0 (mod m), then gcd(a, m) = a, and a won't even have a modular multiplicative inverse. (wikipedia.org)
When ax ≡ 1 (mod m) has a solution it is often denoted in this way − x ≡ a − 1 ( mod m ) , {\displaystyle x\equiv a^{-1}{\pmod {m}},} but this can be considered an abuse of notation since it could be misinterpreted as the reciprocal of a {\displaystyle a} (which, contrary to the modular multiplicative inverse, is not an integer except when a is 1 or -1). (wikipedia.org)
Multiplication3
The notation would be proper if a is interpreted as a token standing for the congruence class a ¯ {\displaystyle {\overline {a}}} , as the multiplicative inverse of a congruence class is a congruence class with the multiplication defined in the next section. (wikipedia.org)
Also, multiplication has one inverse operation: division. (wikipedia.org)
going left one tick mark is division by 10 1/10 , or multiplication by 10 -1/10 ≈ 0.79. (exploringbinary.com)
Modulus1
If a does have an inverse modulo m, then there are an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. (wikipedia.org)
Displaystyle1
In the standard notation of modular arithmetic this congruence is written as a x ≡ 1 ( mod m ) , {\displaystyle ax\equiv 1{\pmod {m}},} which is the shorthand way of writing the statement that m divides (evenly) the quantity ax − 1, or, put another way, the remainder after dividing ax by the integer m is 1. (wikipedia.org)
Difference1
For a given positive integer m, two integers, a and b, are said to be congruent modulo m if m divides their difference. (wikipedia.org)