Examples: 15 = 3 (mod 2) since their difference 12 is divisible by 2. They are also congruent mod 3, mod 4, mod 6 and mod 12. Any two odd numbers are congruent mod 2. Any two even numbers are congruent mod 2. 1234 = 4321 (mod 9).
Practically speaking, any integer is congruent mod N to its remainder after division by N. For example, any number is congruent mod 11 to one of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 since these are the only possible remainders.
For practice, here is a table of seventh powers (mod 11).
x x^{7} (mod 11) 0 0 1 1 2 128 = 7 (mod 11) 3 2187 = 9 (mod 11) 4 16384 = 5 (mod 11) 5 78125 = 3 (mod 11) 6 279936 = 8 (mod 11) 7 823543 = 6 (mod 11) 8 2097152 = 2 (mod 11) 9 4782969 = 4 (mod 11) 10 10000000 = 10 (mod 11)
Comments: webmaster@ams.org
© copyright 1999, American Mathematical Society.