verify and explain

Question

Let a = 67, b = 30, and n = 7
Verify and explain that (a-n)≡(mod b)

Answer 1

a = 67, b = 30, and n = 7.

Verify and explain that (a - n) ≡ (mod b).

 

(67 - 7) ≡? (mod 30)

 

For this to be true, 67 - 7 must be an integer multiple of 30 (see Wolfram's ModularArithmetic or Wikipedia's Modular_arithmetic#Congruence_relation).

 

And it is: 67 - 7 = 60 = 2*30. Q.E.D.

Answer 2

( a - n) ≡  (mod b)

 

 a ( mod b )- n ( mod b) =

 

67 ( mod 30) - 7 ( modb)=

 

       7   - 0 = 7