Question #6d78f

1 Answer
sente
Dec 3, 2016

#2*4-=1" (mod "7")"#, thus #4=2^(-1)# in #ZZ_7#.

Explanation:

(This answer assumes a basic understanding of modular arithmetic)

When working in #ZZ_n#, if #k in ZZ_n# is coprime with #n#, that is, if #"GCD"(k, n) = 1# then there exists a unique #k^(-1) in ZZ_n# such that #k*k^(-1) -= 1" (mod "n")"#. We call #k^(-1)# the multiplicative inverse of #k# in #ZZ_n#.

To see that #2^(-1) -= 4# in #ZZ_7#, then, we need only show that #2*4 -= 1" (mod "7")"#. Indeed,

#2*4-=8-=1+7-=1" (mod "7")"#

Thus #2^(-1) = 4# in #ZZ_7#

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