Modular Equivalence

What is Modular Equivalence?

Two integers are said to be modular equivalent if they have the same remainder in modular arithmetic.

Example 5.1.1:

Show that the following pairs of numbers are equivalent in the moduli given: 

(i) 67 and 94 (mod 9)
(ii) 108 and 97 (mod 11)

Solution

(i) 67 and 94 (mod 9)

67 = 7 × 9 + 4

94 = 10 × 9 + 4

Note that the remainder is 4 in each case.

Thus  \( \scriptsize 94 \equiv 67 \:( mod \:9) \)

The integers 67 and 94 are congruent because they have the same modular equivalent

(ii) 108 and 97 (mod 11)

Thus  \( \scriptsize 97 \equiv 108 \:( mod\: 11) \)

Hint:
In general, two integers x and y  are said to be congruent modulo z, i.e. 
\( \scriptsize x \equiv y \:( mod\: z) \)

i.e. If the difference of x – y is an integer multiple of z, where the number z is referred to as the modulus of the congruence.