Two integers are said to be modular equivalent if they have the same remainder in modular arithmetic.
Show that the following pairs of numbers are equivalent in the moduli given:
(i) 67 and 94 (mod9) (ii) 108 and 97 (mod11)
67 = 7 x 9 + 4
94 = 10 x 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)\( \scriptsize 108 \equiv 9 \: \times \: 11 \: + \: 9 \) \(\scriptsize 97 \equiv 8 \: \times \: 11 \: + \: 9 \)
Thus \( \scriptsize 97 \equiv 108 ( mod 11) \)
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.