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For any integers a and b, and positive integer n, consider the following statement: Statement: 1 If a ≡ b mod n and c ≡ d mod n then a + c ≡ b + d mod n. Statement: 2 If a ≡ b mod n, and c is a positive integer, then ca ≡ cb mod cn Statement: 3 If ab ≡ ac mod n and if gcd(a, n) = 1, then we have b ≡ c mod n. Which of the following statement is/are correct.

For any integers a and b, and positive integer n, consider the following statement: Statement: 1 If a ≡ b mod n and c ≡ d mod n then a + c ≡ b + d mod n. Statement: 2 If a ≡ b mod n, and c is a positive integer, then ca ≡ cb mod cn Statement: 3 If ab ≡ ac mod n and if gcd(a, n) = 1, then we have b ≡ c mod n. Which of the following statement is/are correct. Correct Answer (1), (2) & (3)

Concept:

1. If a ≡ b mod n and c ≡ d mod n then,

  • a + c ≡ b + d mod n.
  • ac ≡ bd mod n

2. If a ≡ b mod n, and c is a positive integer, then ca ≡ cb mod cn

3. If ab ≡ ac mod n and if gcd(a, n) = 1, then we have b ≡ c mod n.

Explanation:

As discussed above, we can see that,

Statement 1, 2 & 3 are correct. 

Important Points1. For any integers a and b, and positive integer n, we have:

  •  a  ≡ a mod n.
  •  If a ≡ b mod n then b ≡ a mod n.
  •  If a ≡ b mod n and b  ≡ c mod n then a ≡ c mod n

2.  If a ≡ b mod n then b = a + nq for some integer q, and conversely.

3. If a ≡ b mod n then a and b leave the same remainder when divided by n. 

4. If gcd(a, n) = 1, then the congruence ax ≡ b mod n has a solution x = c. In this case, the general solution of the congruence is given by x ≡ c mod n.

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