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

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: Assume that k mod 15 4. Compute 11k mod 15. Show all your work clearly.

A: Topic:- basic maths

Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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: (a) Show that if n 7 mod 8, then n + 6n 3 mod 8.

A: If, n = 7 (mod 8) So, n² = 7² (mod 8) So, n² = 49 (mod 8) So, n² = 1 (mod 8)

Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

A:

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: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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Q: He have to Show that lon = (-1)" (mod 1") shall use mathematical moluction Let Pin) := lon a (47"…

A:

Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

A:

Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

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

A:

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

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: 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: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: Establish that if a is an odd integer, then 2" а = 1 (mod 2" +2) for any n 2 1.

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: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

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

A:

Q: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

A:

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

A:

Q: . How many different values are there for 4" (mod 7), as n ranges over all positive integers?

A: Note - As per our guidelines we have to answer first question if multiple questions are posted. So…

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

A:

Q: 25 Prove that mod 7 5 for all Gindegars and bin b mod 7-6

A:

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

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Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: Q3/ Use Chinese theorem to solve the following congruence: X=2 mod 3 X=1 mod 5 X 2 mod 4

A: Since you have posted a multiple question ,I will solve the first question(i.e.Q3)  for you. To get…

Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

A: Suppose P(n) be any mathematical statement in terms of n. Then Principle of mathematical induction…

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

A:

Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

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: 10. Find the sum of all incongruent solutions of a =1 (mod 17). (Enter the least non-negative…

A:

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: (a) gcd(q, n) = 1 if and only if kq # 0 mod (n) for all 1 < k< n. %3D

A:

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: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.