Question

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.

a.(1) and (2)
b.(1) and (3)
c.Only (2)
d.(1), (2) & (3)

Answer 1

The correct statements are (1) and (3). Statement 1 states that if a ≡ b mod n and c ≡ d mod n, then a + c ≡ b + d mod n. Statement 3 states that if ab ≡ ac mod n and gcd(a, n) = 1, then we have b ≡ c mod n.

Step-by-step explanation:

The correct statements are (1) and (3).

Statement 1 states that if a ≡ b mod n and c ≡ d mod n, then a + c ≡ b + d mod n. This can be proved by using the properties of congruence and modular arithmetic. For example, if a ≡ b mod n, then a = b + kn, where k is an integer. Similarly, c = d + ln, where l is an integer. Then a + c = (b + kn) + (d + ln) = (b + d) + (k + l)n, which implies that a + c ≡ b + d mod n.

Statement 3 states that if ab ≡ ac mod n and gcd(a, n) = 1, then we have b ≡ c mod n. This can be proved using the cancellation property of congruence. If ab ≡ ac mod n, then ab - ac = a(b - c) is divisible by n. Since gcd(a, n) = 1, it follows that n must divide b - c, which implies that b ≡ c mod n.