I need help with modular arithmetic

Question

Compute 5ˆ250 mod 26,
clearly outlining all steps in the process.

Answer 1

There is a pattern of remainders when dividing 2ⁿ/26:

 

n [exponent of 5]                              1     2     3     4     5     6     7     8     9     10    11   12

Remainder [that is, 5ⁿ MOD 26]:    5     25   21    1     5    25   21    1     5     25    21    1     

 

So, what is the remainder when n=250 ?

  250/4 is 62 remainder 2    [The pattern repeats every four numbers]

 

  5²⁴⁸ MOD 26 is 1      [248 is a multiple of 4]

  5²⁴⁹ MOD 26 is 5

  5²⁵⁰ MOD 26 is 25