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

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…

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Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: 3. Prove that ox + 6x -4X12 is irreducible over Q.

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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…

A: Your questions solution below in images

Q: 1. Let a, b, c, r, s, and t be integers with r>s>t>1. If a = b (mod r), a = c (mod s), and gcd(r,s)…

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

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Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

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Q: Show that (p-1)(p2) (pr) (-1)'r!(mod p), for r= 1, 2, . . . ,p - 1. .

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Q: 2. If a = 3 (mod 11) and b = 9 (mod 11), compute the values, positive and fully-reduced modulo the…

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Q: 1. Find all r E Z satisfying each of the following equations. (a) 3r = 2 (mod 7) (d) 9r = 3 (mod 5)…

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Q: (6) Compute ind2 (19) modulo 101.

A: We recall the definition of a primitive root.

Q: Find all 6 permutations of (12)(345).

A: Find all permutations with 12 in the first pair of bracket : (12)(345) (12)(354) (12)(435) (12)(453)…

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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

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Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: 4. Using your work from the earlier parts or independently find: (a) mod 8 (b) + mod 23 17 (c) mod 7

A: Given

Q: Find the amount of money owed at the end of 5 years if $60,000 is borrowed at 7% per year compounded…

A: Here we have to find amount of money at the end of 5 years if money borrowed is $60, 000 at 7% rate…

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)…

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Q: *1. What is the least residue of 56 (mod 7) 58 (mod 7) 19458 (mod 7)?

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).

A: In Solution we use divisibility formula that  If a and b are integers, a divides b if there is an…

Q: 9. Prove that if n = 3 (mod 4), then the equation x² + y² = n has no integer solution.

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