Suppose that a1 ≡ b1 mod (m) and a2 ≡ b2 mod (m) -Show that a k 1 ≡ b k 1 mod (m) (hint: use induction)

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

A:

Q: Let be P odd an prime s.t- gcd (y² + 4v₁ p) = 1. 2 If (x + y √B ) ² = u ² + 4v (mod pk) (VK 6 N),…

A:

Q: If a cy (mod n) and c> 0 then a) x y (mod en) b) x y (mod n) ) x=y (mod) d) None of these :Select…

A:

Q: 5. If ged(a, n) = 1 then the order of a in (Z/nZ)× is the smallest integer i such that a¹ = 1( mod…

A: Since gcd(a, n) = 1, then             aφ(n) ≡ 1 (mod n)  Now, let order(a) = k.        => by…

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

A:

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

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: Given that b₁ = 3, b2 = 5 and b=3b-1-2b-2 for all integers n 23, use strong induction to prove that…

A: Given b1 = 3 ,b2 = 5 , and bn = 3bn-1 - 2bn-2 for all integers n ≥ 3.We will prove by mathematical…

Q: Prove that's is prime p, then (a + b)² = (a² + b²)(mod p), Va; b € Z

A:

Q: 4. Let a, b, c, d, n € Z such that a = b (mod n) and c= d (mod n). (a) Prove that a+c=b+d (mod n).…

A:

Q: If x = y (mod n) and u = v (mod n) then which of the following is true (a) x + u = y + v (mod n) (b)…

A: Given that x≡y (mod n) and u≡v (mod n)

Q: Prove that if x² = y² (mod n) and x #±y (mod n) then gcd(x + y, n) is a nontrivial factor of n.

A: We have,x2≡y2mod nAnd,x≢±ymod n. We have to show that gcdx+y, n is a nontrivial factor of n.

Q: 11. Prove that a¹ = 1 (mod 5) for every integer a that is not divisible by 5.

A: We have to prove  for every integer  that is not divisible by

Q: Exercise 16. Prove (without induction) that B" = 0 mod B for all positive integers B and n. Hint:…

A: Congruence and proof without induction

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: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers 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: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

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: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

A:

Q: 2. Prove that if a = b( mod r) and c = d( mod r) then a – c= b – d( mod r)

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: *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: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

A:

Q: Prove that ifa≡b(modn)andb≡c(modn), thena≡c(modn). Note that you must use thedivides definition of…

A:

Q: 13. Suppose that z1 V3 +i and z2 = 2/2+ 2v2i. 22 (a) Write z1 and z2 in mod-arg form. (b) Hence…

A:

Q: Compute the following: (your answer should be the least nonnegative solution). Show reasoning (i) 87…

A:

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

A:

Q: Show that (a + p)P = a (mod p2), where p is prime, and that, more generally, a = b (mod p) ⇒a= b…

A: To prove that (a+p)p≡ap(modp2), where p is a prime, it can use binomial expansion and Fermat's…

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

A:

Q: Apply the properties of congruence to make computations in modulo n feasible. Find the unique r such…

A:

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 know that the property of congruence

Q: Let n1, n2, n3 be three numbers that are pairwise relatively prime so that (n1, n2) = (n2, n3) =…

A: Given that n1,n2 and n3 are three numbers that are pairwise relatively prime. We have to prove…

Q: 1. Let m be a positive integer. If a = b (mod m) and c= d (mod m), a) Prove a+c= b + d (mod m) b)…

A: