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Is this approach right in proving a theorem 

Ques: show that a mod m = b mod m if a is congruent to b (mod m)

Proof: given a is congruent to b(mod m)

According to definition: 
a - b / m 
i.e a - b = mx  (for some integer x).     ....(1)

Also    a = mx + b
that can also be written as  a mod m = b  ...(2)

Also again from equation (1)

-b = mx - a
b = -mx + a
Or b = m(-x) + a
Or b = my + a    ...(3)

Therefore b mod m = a 

If we replace b in equation (2) by equation (3) then we get 

a mod m = my + a 
my + a can also be written as b mod m 

Therefore, a mod m = b mod m 

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