if a ≡ b (mod n then a and b have the same remainder when divided by n)

Prove that a ≡ b (mod n) if and only if a and b leave the same remainder when divided by n Proof ⇒ Suppose a ≡ b (mod n) Then, by definition, we have

Proposition 3 Two integers a, b are congruent mod n if and only if they have the same remainder when divided by n Proof First 

First we notice that if p divides both a and b then p2 divides both a2 and b2 Prove that a ≡ b (mod n if and only if a and b have the same remainder mod n

Modular arithmetic is the “arithmetic of remainders ” The somewhat b Examples We have: 7 ≡ 22 (mod 5), −4 ≡ 3 (mod 7), 19 ≡ 119 Let n ∈ N and a,b ∈ Z Then a ≡ b (mod n) ifi a and b leave the same remainder when divided by n

12 nov 2014 · natural number m if and only if they have the same only if they have the same remainder upon division by m E g 3 ≡ 7 (mod 2) 9 ≡ 99 Let a, b ∈ ℤ, m ∈ ℕ a and b are said to be congruent modulo m, written a ≡ b (mod m), if and only if a – b If a ≡ b (mod m) and c ≡ d (mod m), then – a + c ≡ b + 

Note: I have attempted to restore as much of the fonts as I could, unfortunately I integers that leave the same remainder when divided by a particular integer third number m, then we say "a is congruent to b modulo m", and write a ≡ b Example 4: Prove that 2 5n + 1 + 5 n + 2 is divisible by 27 for any positive integer n

of a then a = mb for some n ∈ Z In our current discussion, to say b is a divisor a ≡ b mod(n) if a and b have the same remainder when divided by n Proof: Suppose a ≡ b mod(n) then a and b share the same remainder after division by n

For example if a ≡ b mod n and b ≡ c mod n, then n (b − a) and n (c − a) But if n modulo n if any only if they have the same remainder with divided by n

If the remainder in the division of n by m is 0, then we say that n is divisible by this case, are all integers that have the same remainder as x when divided by m) that, if x≡x' (mod m) and y≡y' (mod m), then there are integers a and b such