2. Find the least nonnegative residue modulo m for each integer n below: (a) n= 100!, m= 101 (b) n = 85!, m= 89 (c) n = 261 m = 13 3. Let p be an odd prime. Show 2(p - 3)! –1 mod p

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: True or false: If p is an odd prime, then (p-1)! + 2^(p-1) congruent 0(mod p). Justify your answer.

A: p is an odd prime.

Q: Let p be an odd prime. Prove that (a) If p = 1 (mod 5) or p = 4 (mod 5), then 5 is quadratic residue…

A: Given p is an odd prime. We have to prove (a). if p≡1(mod 5)  or  p≡4(mod 5), then 5 is 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: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: Prove with induction the sum of first n Fibonacci numbers with even indices, i.e., f2 +f4 +···+f2n…

A:

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: 7. Let n > 1. Suppose that a"-1 = for all non-zero remainders a mod n. Show that n is prime. an-1 =…

A: Given that, n≥1 and an−1≡1modn We have to show that n is prime. We know that Euler's generalization…

Q: If x108 = 1(mod 109) then possible value of x is (а) 108 (b) 218 (c) 109 (d) None c Select one: • bx…

A: Find the possible value of x If x108= 1(mod109).

Q: Consider the sequence an = in + (-i). Even though the formula for a refers to complex numbers, all…

A:

Q: What is the largest value that m mod n can be?

A: Largest value of m mod n is n-1.

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

A:

Q: (H2) Determine the number of integers n < 1000 that have primitive element modulo n. a

A: Solution: We will use the following fact: The only numbers that admit a primitive root are those of…

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: Let p, q, and r be prime natural numbers with p # q and n= p2q. Then, all the subfields of GF(r")…

A: Answer: Option (C) Given that, p,qand r are prime natural numbers and n is positive integer. Let F…

Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Suppose that n is an integer such that n≥3. Using a combinatorial proof, show that:

A:

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

A: We know that,

Q: Show that II (1-) < II p° perfect = n = p*||n p*||n Hint: The following formula should prove useful:…

A:

Q: Prove a2 = c² (mod n), assuming that a, c and n are integers with n > 1 and that a = c (mod n)

A:

Q: Let F(n) denote the Fibonacci numbers. Prove the following property for n ≥ 3 directly from the…

A:

Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: Show that if n is an odd perfect numb prime and pa = 1 (mod 4). Use part (a) to show that if n n = 1…

A: Here in the question, We have to show n is an odd perfect number then, n = pαm2 where, p is an odd…

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

A:

Q: (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),…

A:

Q: Show that 32n 1 is always a multiple of 8 for all n = N = {1, 2, 3,...}. We say one integer A is a…

A: Step 1:Step 2:Shows a proof by induction that 32n - 1 is always a multiple of 8 for all natural…

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: 53!=r(mod61)

A:

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

A:

Q: Show that 10n <3n for n ≥no. Be sure to find no first.

A:

Q: Determine the following. 1. AnB 2. AUB 3. |A|

A:

Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

A:

Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

A:

Q: (24) Let p, q € N be two distinct primes. (a) Find the following quantities (possibly in terms of p…

A: Given: p, q∈ℕ are two distinct primes. To Do: (a) Find gcdp, q,   gcdp2q, pq2,   gcdpq, p2q2+1. (b)…

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Q: (14.)(a) Prove or disprove: If ab= 0 (mod n), then a 0 (mod n) or b = 0 (mod n). (b) Do part (a)…

A:

Q: Use appropriate properties of congruence modulo m to find the least nonnegative residue of n modulo…

A: Recall eulers theorem.