Modular Exponentiation

Example 1

$23^3\mathrm{mod}30$
This equation can also be represented as $-7^3\mathrm{mod}30$
= $-7^2\ast -7\mathrm{mod}30$
= $-133\mathrm{mod}30$
= $17\mathrm{mod}30$
= 17

Example 2

When the difference between the numbers is very large we can find the modulus values for the factors of the power and then multiply them all together

$88^7\mathrm{mod}187$

= $88^1\mathrm{mod}187=88$
= $88^2\mathrm{mod}187=7744\mathrm{mod}187=77$
= $88^4\mathrm{mod}187=5929\mathrm{mod}187=132$

= $88^7\mathrm{mod}187=88^4\ast 88^2\ast 88^1\mathrm{mod}187=894432\mathrm{mod}187$
= 11