300 element multiplicative group mod 341. Examples n = 9 The smallest example with a nontrivial subgroup of false witnesses is . There are 6 residues coprime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to , it follows that 88 is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime). These are in fact the only ones, so the subgroup {1,8} is the subgroup of false witnesses. The same argument shows that is a "false witness" for any odd composite n. n = 91 For n = 91 (= 7 × 13), there are residues coprime to 91, half of them (i.e., 36 of them) are false witnesses of 91, namely 1, 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, and 90, since for these values of x, x90 is congruent to 1 mod 91. n = 561 n = 561 (= 3 × 11 × 17) is a Carmichael number, thus s560 is congruent to 1 modulo 561 for any integer s coprime to 561. The subgroup of false witnesses is, in this case, not proper; it is the entire group of multiplicative units modulo 561, which consists of 320 residues. Examples This table shows the cyclic decomposition of and a generating set for n ≤ 128. The decomposition and generating sets are not unique; for example, (but ). The table below lists the shortest decomposition (among those, the lexicographically first is chosen – this guarantees isomorphic groups are listed with the same decompositions). The generating set is also chosen to be as short as possible, and for n with primitive root, the smallest primitive root modulo n is listed. For example, take . Then means that the order of the group is 8 (i.e., there are 8 numbers less than 20 and coprime to it); means the order of each element divides 4, that is, the fourth power of any number coprime to 20 is congruent to 1 (mod 20). The set {3,19} generates the group, which means that every element of is of the form (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly