8. Use Euler's Theorem to find a number a between 0 and 9 such that a is congruent to 7 1000 modulo 10. (Note that this is the same as the last digit of the decimal expansion of 7 1000,)

Q: The number 73 is prime. Enter an a such that 0 <a < 73 and a = 6 (mod 73), if such an a exists.…

A:

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a)…

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

Q: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

Q: Suppose a is an integer that is relatively prime to 400. Prove that a40 = 1 (mod 400).

A:

Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

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

A: Please see the attachment

Q: Find 96307 7 (mod 739) using the techniques described in Example 8.4.4 and Example 8.4.5. First…

A: To find 96307mod739. It can be computed by evaluating the following,…

Q: to show that the numbers in question are incongruent modulo n.] 11. Verify that 0, 1, 2, 22, 23,…

A: We shall solve this question in next step.

Q: Recall that 2027 is a prime number. Compute all values of x E {0, 1, 2, ..., 2026} such that 999x =…

A:

Q: It can be shown that every integer can be uniquely represented in the form ek1341 where e=-1, 0, or…

A: We need to find balanced ternary expansion of 11 . That means , we need to find  the expansion of 11…

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: The Modular Operation r mod m = r denotes that r is the remainder = 3. If two integers have the same…

A: Rewrite the following theorems using quantifiers and predicates.

Q: Prove that for any integer k, at least one of k, k + 2, or k + 4 is divisible by 3. (Hint: consider…

A:

Q: prove that the following integer is composite (not prime). Do not use mod. 2^35(2^33-1) + 1

A:

Q: 12. Prove that there are infinitely many prime numbers in the arith- metic progression 1,4,7, 10,…

A:

Q: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

A:

Q: 8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b

A:

Q: 3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.

A: Wilson's Theorem: Let k be a prime number, then by Wilson's theorem, (k-1)!=-1(modk)

Q: Number theory

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: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

A:

Q: (2) Use the power property of modular arithmetic and a technique discussed in section 8.4 to…

A:

Q: 2

A:

Q: 5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 ible by 7.

A:

Q: 1. Show that an integer n is divisible by 2 if and only if its last digit is even. (Hint: For each i…

A:

Q: Consider a set of integers from 1 - 10. G = {1,2,3,4,5,6,7,8,9,10} Answer the following questions:…

A: (i). Given G = {1,2,3,4,5,6,7,8,9,10}

Q: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

A:

Q: Consider the congruence mod 6 for integers. Z6={[0], [1], [2], [3], [4], [5]}. Prove the product of…

A:

Q: Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if the division of c…

A:

Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

Q: Find the modular residue of the product. 9,512,667 2,183,321 (mod 5) 9,512,667 2,183,321 (mod 5) =…

A:

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

A:

Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: When we work with a congruence class, we have to keep two key things in mind. First, we have to know…

A: For ℤ3, The ones that are congruent to 0 is the set {...,-6,-3,0,3,6,..}. We call it the congruence…

Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: Suppose that a, b, m, and n are integers (n, m positive) and suppose we know a = b (mod n) and n | m…

A: We first show how the conclusion is true from that we choose the right option.

Q: a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the…

A:

Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

A:

Q: Prove that if n is an odd integer, then n^2 is congruent to 1 modulo 8.

A:

Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

A:

Q: Show that : If n is an integer then n is congruent to 0 modulo 4 or 1 modulo 4.

A:

Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: 13. |Using the definition of modular arithmetic, prove that if a =b mod n and c = d mod n, then ac =…

A:

Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

A:

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: (a) Let n be a positive integer. Prove that n is congruent to the sum of its digits modulo 9. Hint:…

A: (a) First we will prove that 9(10m-1) for every m∈ℕ : Since, 10≡1mod 9 , we can write this as  910-1…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.