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What is the step-by-step algorithmic process to find the modular multiplicative inverse of (n-1) mod np, where (n-1) and np are not coprime?

What you're asking for is impossible. You're looking for x such that (n-1)x = 1 mod np. However, both n-1 and np are even. It's easy to see that any multiple of n-1 will be even. Thus, any multiple of ...

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