Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n = (j+ k) mod n.

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

A:

Q: Find an integer n such that n = 1 mod 7, n = 3 mod 8, and n = 2 mod 9.

A: We can solve this system of congruences using the Chinese Remainder Theorem.

Q: if a equiv b (mod n) , then a⁴ equiv b⁴ (mod n) for any positive integer 4.

A:

Q: If a = b (mod m) and a = b (mod n) then a = b (mod mn)

A:

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: (b) Prove that if a = b (mod m) and if b = c (mod m), then a = c (mod m).

A:

Q: If a =b (mod m) and a = b (mod n) then a = b (mod m+n) Select one: O True O False

A:

Q: If a ≡ b (mod n1) and a ≡ b (mod n2), must it be true that a ≡ b (mod n1 + n2)? Either give a proof…

A:

Q: If a ≡ b (mod n1) and a ≡ b (mod n2), must it be true that a ≡ b (mod n1n2)? What if gcd(n1, n2) =…

A: If a≡b(modn1) and a≡b(modn2), it is not necessarily true that a≡b(modn1n2). The statement is true…

Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

A:

Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

A:

Q: Prove that if a = b (mod m) and if c = d (mod m), then a+c=b+d (mod m).

A:

Q: Prove that if a = b (mod m) and if c = d (mod m), then ac = bd (mod m).

A: a≡b   (mod m)⇒ m|(a−b)That is there exist an integer k such that a−b=mk

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

A:

Q: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

A:

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

A:

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

A:

Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: State whether True or False: If A and B are integers and A ≡ 3 (mod 3) and B ≡ 2 (mod 6) then A+B ≡…

A:

Q: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

A:

Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: (f) If a = b (mod n), then a* bk (mod n) for any positive integer k.

A:

Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

A:

Q: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: 4. Prove that if a = b (modn) and m|n, then a = b (modm)…....

A: