(5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B), then n² = 1 (mod 12)

Q: Let a, m € Z+ and (a,m) = (a - 1, m) = 1. Then prove that 1+a+...+ a(m)-1 = 0(modm)

A: For given,       (a,m) = (a -1, m) = 1 We have to show,      1 + a + a 2 + ........ + a φ (m)-1 ≡ 0…

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

A:

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

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: The remainder when a is divided by 7 is 2 and the remainder when b is divided by 7 is 5. (So a mod 7…

A: Given that remainder when "a" is divided by 7 is 2

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

A:

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b3 c (тod n) (b) b2 =…

A: Given below the detailed solution

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: Q3 Not submitted Section 4.1 # 14 Show that if a, b, and e are integers with e > 0 such that a = b…

A:

Q: Let a, n be positive integers. Knowing that 15a – 13n = 1, we deduce that the multiplicative inverse…

A: We have to find the moduler multiplicative inverse of a.

Q: (c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n

A: We know that,

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

A:

Q: 2. Let f: Z-Z be given by the following rule: x+6 x=0 x = 1 x+8 x 2 f(x)=x+5 (mod 3) (mod 3) (mod 3)…

A: The given function is, fx=x+6x≡0 mod3x+5x≡1 mod3x+8x≡2 mod3 To Find: (a) Example to show that f is…

Q: (a) Prove that 318 = q (mod 19).

A:

Q: If a =b (mod m) then a" = b" (modm) Vn e Z*.

A: Given a≡bmod m. we have to show that an≡bnmod m. we wil prove this by result by using method of…

Q: Compute (1001 mod d) for d = 2, 3,..., 10.

A:

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

A:

Q: (7) Let m≥ 1 and let R= {0, 1, 2,...,m - 1}. (a) Prove that if r1, T2 € R and r₁ r2 (mod m), then r₁…

A:

Q: Let A = {1, 2, ..., 12}. Let aRb mean a = b (mod 5). (a) Give [2]R. [3]R, and [5]R. (b) Give [2]…

A: A set of ordered pairs is referred to as a relation in the area of mathematics and computer science.…

Q: Let p be an odd prime number. Prove the following statements. (a) (7) = 1 if and only if p = 1, 3…

A: ANSWER

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

A:

Q: Proposition 1.2. Say a = a' (mod m) and b = b' (mod m), then (i) a+b=a'+b' (mod m), and (ii) ab a'b'…

A: x=y (mod n)⇒ n|(x−y)That is  x−y=nk  for some integer k

Q: please do 7

A: (a) p is a prime number. Since,

Q: ove that n’ +5n is divisible by 6 for all n E N.

A: