Prove that for all integers m and n,m=n (mod 3) if, and only if,
Prove that for all integers m and n any positive integer if, and only if, .
For all integers m and n, m = n ( mod 3 ) if, and only if, m mod 3 = n mod 3.
m and n, are integers.
Let m and n be integers.
Let us assume (mod 3). By the definition of m being congruent to n modulo 3:
By the definition of divides, there exists an integer k such that:
Add n to each side:
Take modulo 3 of each side of the previous equation:
We can simplify the right side, since
For all integers m and n and any positive integer d, m = n (mod d ) if, and only if, m mod d = n mod d.
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