Let n be a positive integer. We write x; for the ith digit of n. (I.e., if n = 217, then x1 = 7, x2 = 1 and x3 = 2.) In particular, n = x1 + x210+ + xk10k-1. Prove the following: (a) n = x1 + x2+ ...+ xk( mod 9). (b) n = x1 – x2 +...+ (-1)k-lxk( mod 11).

Q: 9. Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

A:

Q: 2p a) n! = 2 (mod p), where for 0 <k < n. %3D k!(n - k)!

A: To prove the given part (a) 2pp≡2mod p where nk=n!k!n-k! for 0≤k≤n

Q: Find an integer x satisfying the equation: 4x21 = 2(mod 5) or show that no such integer exists.

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: The integer n = 12700140x3243243 is missing the digit x. a) If n congruent to  3 (mod 9), then what…

A: According to the given information, Consider the given integer n:

Q: Let x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

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

Q: Suppose that 2n−1 ≡ 1 (mod n). Is n necessarily prime?

A:

Q: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

Q: Suppose b is any integer. If b mod 12 = 7, what is 8b mod 12? In other words, if division of b by 12…

A: Given b mod 12= 7  To find 8b mod 12

Q: The notation x = y mod n is used to indicate the unique integer x in the range 0 :$ x &lt; n such…

A: Given statement is that the notation x = y mod n is used to indicate the unique integer x in the…

Q: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

Q: Use the table in VII to give a formal proof of the following. For all integers a and b, if a = 1…

A:

Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

A:

Q: For D = {1, 2, 3, 4, ..., 999}, find a counterexample to the claim, "For all n E D, if n mod 4 = 2…

A:

Q: Suppose b is any integer., If b mod 12 = 7, what is 4b mod 12? In other words, if division of b by…

A: Given : b mod 12 = 7

Q: Prove, by induction, that, for n2 5, the nth Fibonacci number satisfies Fn s 2".

A:

Q: prove, for all integers x, if x mod 3 does not equal 0, then 3 | (x2 +2)

A:

Q: For which positive values of k is nk = n (mod 7) for all integers n?

A:

Q: Suppose b is any integer. If  b mod 12 = 5,  what is 6b mod 12? In other words, if division of b by…

A:

Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

A:

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

A:

Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

A:

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Prove the following results. (I). If n = 2(mod 4), then n is the sum of three squares of integers.…

A:

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

A:

Q: Prove the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

Q: If a has order n-1 mod n, then n is prime. O True O False

A:

Q: Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

A:

Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

Q: Prove that if n is an odd positive integer, then n² = 1 (mod 8).

A:

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

A:

Q: A sequence of pseudorandom numbers is generated as follows Xo = 4 x₁ = ( 6x₁-1 + 5 ) mod 13 if i > 0…

A:

Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: Deduce the following statements: i) an odd prime number p is the sum of two squares of integers if…

A:

Q: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

Q: For D = {1, 2, 3, 4, ..., 999}, find a counterexample to the claim, "For all n e D, if n mod 3 =1…

A: Given D=1, 2, 3, 4, ..., 999 Here, we have to find a counterexample to the claim,  ∀n∈D, if n mod…

Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

A:

Q: Prove, for all integers x, if x mod 3 is not 0, then 3|(x2+2).

A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let n be any fixed positive integer. Then…

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 that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.

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

A: Looking at the definition of congurence we can easily do these two problems and we need some algebra…

Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

A:

Q: Let k be an integer such that k = -1 (mod 5). What is the remainder when 50k10 + 26k³ – 17 is…

A: Since you have asked multiple questions, we will solve the first Question  for you. If youwant any…