Compute the least residue of (359 - 6)! modulo 359. Note that 359 is prime.

Q: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

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Q: Establish by mathematical induction that any postage of at least 12c can be obtained using 3c and 7c…

A: We have to establish by mathematical induction that any postage of at least 12c can be obtainedusing…

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: Name all integers from -100 to 100 that are congruent to -1 modulo 25!

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Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: Determine the least residue of 56^1010 (mod 323).

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Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

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Q: 2. Find the least nonnegative residue modulo m for each integer n below: (a) n= 100!, m= 101 (b) n =…

A: We have to find the least nonnegative residue modulo m for each integer n below : (a) n=100!, m=101…

Q: Find the least residue of 448 modulo 7

A: Given problem is : Find the least residue of 448 modulo 7

Q: Find the least residue of  -5 -46 modulo 14.

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Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

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

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Q: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

A: The Fibonacci numbers are the numbers in the following integer sequence.…

Q: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

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: 122 is congruent to 5 modulo17 ?

A: To check whether -122 is congruent to 5 modulo 17 or not

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: Show that 18! ≡ -1 (mod 437).

A: We have to show that: 18!≡-1mod437

Q: Let c be an element of N, and p be a prime. Prove that if c is greater than or equal to 3, then c^p…

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Q: Find the order of 4 modulo 49.

A: Evaluate order of 4 modulo 49.

Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

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Q: show that no integer of the form n^3+1 is a prime other than 2

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Q: How many positive integers with 6 or less digits are odd or divisible by 5 and do not contain any…

A: We can solve this problem by considering two cases:

Q: show that 10 is a perfect square mod 45 and give at least two square roots (between 0 and 44)

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Q: Use the PMI to prove that 6" – 1 is divisible by 5 for every integer n > 0.

A: To prove that 6^n-1 is divisible by 5 for every integer greater that or equals to 0.

Q: 19. Let N be the positive integer 7777...777, a 313-digit number where cach digit is a 7. Let f(r)…

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Q: Show that the sum of twin primes p and p + 2 is divisible by 12, provided that p> 3.

A: Let p, p + 2 be twin primes. Then p *(p+1) + 1 = p^2 + 2p + 1 = (p+1)^2 ,  a perfect square.

Q: Show that if p = 1 (mod 4), then x² = -1 (mod p) has a solution given by the least residue (mod p)…

A: Given: p≡1 mod 4We need to prove that x2≡-1 mod p has a solution given by the least residue mod p of…

Q: Find all solutions of 3* = 17 (mod 224)

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Q: For every prime p, show that ( p - 2)! mod p = 1.

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Q: Prove that 7has no expression as an integer liner combination of 18209 and 19043

A: To show that 7 has no expression as an integer linear combination of 18209 and 19043. That is we…

Q: [Hint: It suffices to show that the numbe 11. Verify that 0, 1, 2, 2², 2³, . , 2° form a complete…

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Q: Simplify 146! (mod 149) to a number in the range {0, 1,... 148}.

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Q: Since the next prime after 7 is​ 11, the smallest composite number whose prime factors are all…

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Q: Prove that the messenger number M19 is a prime hence the integer n = 218 (219 -1) is perfect

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Q: 1. Show that if pla + a + a? + a+1 and a -1 mod 5 then P=1 mod 5. Hint: a -1= (a - 1)(a* +a + a? +…

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Q: For n > 3, prove that the numbers ±5, ±52, · reduced residue system modulo 2". form a

A: A subset R of the integers is called reduced a residue system modulo n if (i) gcdr,n =1 for each r…