Modular multiplicative inverse is used for The Chinese Remainder Theorem and RSA algorithm. You can visit Wikipedia.
Normal Modulus
X = M (mod Y)
You can solve that with M = mod(X,Y)
Inverse Modulus
X.B = M (mod Y)
given X,M,Y calculate B
B = inverse_modulus(X,M,Y)
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The inverse modulus would be to find X such that mod(X,Y) = M where M and Y are known (or X === M (mod Y)); this is the chinese remainder theorem which is generalized for any number of Y's and M's when all have the same X and the GCD of all Y = 1 (greatest common divisor). The author is actually requesting Y*Z + M = X*B, which is not the same thing, or the inverse modulus.