Reduced Residue System/Examples/Modulo 18/Powers of 5

Examples of Reduced Residue Systems

$\set {1, 5, 25, 125, 625, 3125}$

We have:

$\ds 25$

$=$

$\ds 1 \times 18 + 7$

$\ds $

$\equiv$

$\ds 7$

$\ds \pmod {18}$

$\ds 125$

$=$

$\ds 6 \times 18 + 17$

$\ds $

$\equiv$

$\ds 17$

$\ds \pmod {18}$

$\ds 625$

$=$

$\ds 34 \times 18 + 13$

$\ds $

$\equiv$

$\ds 13$

$\ds \pmod {18}$

$\ds 3125$

$=$

$\ds 173 \times 18 + 11$

$\ds $

$\equiv$

$\ds 11$

$\ds \pmod {18}$

Thus we see that:

$\set {1, 5, 25, 125, 625, 3125}$

is equivalent to:

$\set {1, 5, 7, 17, 13, 11}$

$\blacksquare$

1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Exercise $2 \ \text {(a)}$