Recall that a number must be coprime with 10 to have a multiplicative inverse modulo 10. We can easily check that:
1, 3, 7, 9 are coprime with 10, so each of them has a multiplicative inverse modulo 10, which can be computed with the help of the extended Euclid algorithm.
2, 4, 5, 6, 8 are not coprime with 10, so none of them has a multiplicative inverse modulo 10.
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