Example
Suppose we wish to find modular multiplicative inverse x of 3 modulo 11.
This is the same as finding x such that
Working in we find one value of x that satisfies this congruence is 4 because
and there are no other values of x in that satisfy this congruence. Therefore, the modular multiplicative inverse of 3 modulo 11 is 4.
Once we have found the inverse of 3 in, we can find other values of x in that also satisfy the congruence. They may be found by adding multiples of m = 11 to the found inverse. Generalizing, all possible x for this example can be formed from
yielding {...,-18,-7,4,15,26,...}.
Read more about this topic: Modular Multiplicative Inverse
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)