Let n be a positive integer, and assume that a and b are integers such that a ≡ b (mod n). Prove each statement below. (a) If q is an integer, then aq ≡ bq (mod n). (b) If an integer d divides both b and n, then d divides a.

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A: Just use the mentioned definition of congruence modulo.

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A: Given: x mod 9=5 then, 9|x2-2x+3|

Q: 9. Let a and b be integers. Prove that if a =7 (mod 8) and b = 3 (mod 8), then: (a) a + b = 2 (mod…

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Q: If al = b (mod p), then prove that a = b (mod p).

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Q: Let C={x €Z: x =7 mod 9} and let D={x € Z,x = 1 mod 3). a) List five elements of C and five elements…

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Q: Q1 Prove or disprove i. If ac = bc(mod m), then a = b(mod m).

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Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

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Q: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

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A: We will prove this by basic result of congruence

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Q: 1. Prove each of the following assertions: (a) If a = b (mod n) and m | n, then a = b (mod m).

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Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

Q: Prove or disprove: If a = b(mod n²) then a = b(mod n)

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Q: 1 Prove each of the following assertions: (a If a = b (mod n) and m |n, then a = b (mod m). (6) If a…

A: We know that  a ≡ b (mod n) is equivalent to n | (b-a)