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…

A:

Q: - Calculate the orders of 2 (mod 67) and of 2 (mod 73).

A:

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

A:

Q: Given |a,b,c,n €Z, show that, if ac bc (mod n), a = b (mod), d= mdc(c, n). then

A:

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

A:

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: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We use the properties of congruences to answer the given question.

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)…

A:

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

A:

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

A:

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

A:

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

A:

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

A:

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: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

A:

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

A:

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…

A:

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

A:

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: Let A = { {a}, {b}, {c}, {d}, {e} } Determine by extension P(A

A:

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: 5. Let (1 2 3 A = 2 1 4 1 0 2 Find A-lif possible. If A-does not exist, explain why.

A: Given See the handwritten solution

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: *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.

A:

Q: compute: 11 31*

A: Let ⊕:Z37⊕ + Z37⊕  →Z37⊕ be defined as a⊕b=(a+b)mod 37

Q: VI. Determine four values of x that will satisfy the following. 16. x = 4 mod 5 & x = 3 mod 7

A: